{"assistant":{"explanations":[{"calculationRefs":["calc-1"],"text":"9 + 16 = 25: обращение теоремы Пифагора даёт прямой угол против стороны 5."},{"calculationRefs":["calc-2"],"text":"36 меньше 16 + 25: угол против большей стороны острый по II.13."},{"calculationRefs":["calc-3"],"text":"16 больше 4 + 9: угол против большей стороны тупой по II.12."},{"calculationRefs":["calc-4"],"text":"49 не равно 1 + 49 и ни одно из остальных равенств не выполняется: прямого угла нет."},{"calculationRefs":["calc-5"],"text":"Нулевые стороны нарушают неравенство треугольника I.20: фигуры нет."},{"calculationRefs":["calc-6"],"text":"Набор 0, 3, 3 треугольником не является, а вывод о прямом угле требует треугольника: ни подтверждения, ни опровержения."}],"origin":"assistant","summary":"Предложение I.48 обращает теорему Пифагора: если квадрат одной стороны равен сумме квадратов двух других, угол между ними прямой. Предложения II.12 и II.13 дают трихотомию: квадрат больше суммы у тупого угла, меньше у острого. Стороны, не удовлетворяющие неравенству треугольника I.20, треугольника не образуют. Вырожденный набор с нулевой стороной не попадает ни под одно предложение, и право честно молчит."},"calculations":[{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"установлено","missingInputs":[],"truthStatus":"TRUE_ONLY"},"closedEditionRules":[],"derived":["trigonon_estin(Треугольник)","orthogonion(Треугольник)"],"derivedOmitted":0,"evaluation":{"proofGraph":{"nodes":[{"attributes":{"assertion":"urn:mcp:case#fact-1"},"conclusion":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":3},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":4},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":5}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#pleurai"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#trigonon_estin"},"evidence":[],"id":"urn:proof:apply:TrigononEstin:aede226a089a3e3c6e4703def6ebb6193f6cc1a19a686f45cde294caf9255862","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:grc:eukleides:clir:pythagoras#TrigononEstin","sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},"v1":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":3},"v2":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":4},"v3":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":5}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#orthogonion"},"evidence":[],"id":"urn:proof:apply:OrthogonionProsTritei:7056bf2dc23ce7e03a854f4450285b51cc7ddbdca773bdba794502331272362a","kind":"rule_application","premises":["urn:proof:apply:TrigononEstin:aede226a089a3e3c6e4703def6ebb6193f6cc1a19a686f45cde294caf9255862","urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:grc:eukleides:clir:pythagoras#OrthogonionProsTritei","sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},"v1":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":3},"v2":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":4},"v3":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":5}}},{"attributes":{},"conclusion":{"constraint":"urn:grc:eukleides:clir:pythagoras#OrthogonionKaiAmblygonionAsymbata","requirementStatus":"NEITHER","status":"UNDETERMINED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:OrthogonionKaiAmblygonionAsymbata:5b3c3e00c9a668a258798c1ee3b3b07267f645534378f22f47fccf14436efa24","kind":"constraint_check","premises":["urn:proof:apply:OrthogonionProsTritei:7056bf2dc23ce7e03a854f4450285b51cc7ddbdca773bdba794502331272362a"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:grc:eukleides:clir:pythagoras#OrthogonionKaiOukOrthogonionAsymbata","requirementStatus":"NEITHER","status":"UNDETERMINED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:OrthogonionKaiOukOrthogonionAsymbata:6260bd8c1ecf8a49585d04d06414c6c5dc7e0bb851fb49b07666a6f3aba3a047","kind":"constraint_check","premises":["urn:proof:apply:OrthogonionProsTritei:7056bf2dc23ce7e03a854f4450285b51cc7ddbdca773bdba794502331272362a"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:grc:eukleides:clir:pythagoras#TrichotomiaPliris","requirementStatus":"TRUE_ONLY","status":"SATISFIED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:TrichotomiaPliris:f61efd2e9cb2fa92658918100a35130495ac0310b3993be1c50d4ed98fef2507","kind":"constraint_check","premises":["urn:proof:apply:OrthogonionProsTritei:7056bf2dc23ce7e03a854f4450285b51cc7ddbdca773bdba794502331272362a","urn:proof:apply:TrigononEstin:aede226a089a3e3c6e4703def6ebb6193f6cc1a19a686f45cde294caf9255862"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"literal":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#orthogonion"},"truthStatus":"TRUE_ONLY"},"evidence":[],"id":"urn:proof:query:mcp","kind":"query_evaluation","premises":["urn:proof:apply:OrthogonionProsTritei:7056bf2dc23ce7e03a854f4450285b51cc7ddbdca773bdba794502331272362a"],"sourceAnchors":[]}],"proofHash":"sha256:58abac9ccca23c56e925f2e137d82a1fea04d448d4982493d24fea9be374ac22","roots":["urn:proof:constraint:OrthogonionKaiAmblygonionAsymbata:5b3c3e00c9a668a258798c1ee3b3b07267f645534378f22f47fccf14436efa24","urn:proof:constraint:OrthogonionKaiOukOrthogonionAsymbata:6260bd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τὸ αὐτὸ τρίγωνον ὀρθογώνιόν τε εἶναι καὶ ἀμβλυγώνιον"},{"language":"ru","status":"unofficial","text":"инвариант §93.1: прямоугольный и тупоугольный — разные роды (определение I.21), и один треугольник не может быть обоими"}],"package":"urn:grc:eukleides:clir:pythagoras"},{"id":"urn:grc:eukleides:clir:pythagoras#OrthogonionKaiOukOrthogonionAsymbata","kind":"constraint","labels":[{"language":"grc","status":"unofficial","text":"ἀδύνατον τὸ αὐτὸ τρίγωνον ὀρθογώνιόν τε εἶναι καὶ οὐκ ὀρθογώνιον"},{"language":"ru","status":"unofficial","text":"инвариант §93.1: треугольник не может быть одновременно прямоугольным по признаку I.48 и не прямоугольным по контрапозиции I.47"}],"package":"urn:grc:eukleides:clir:pythagoras"},{"id":"urn:grc:eukleides:clir:pythagoras#OrthogonionProsTritei","kind":"rule","labels":[{"language":"grc","status":"official","text":"Ἐὰν τριγώνου τὸ ἀπὸ μιᾶς τῶν πλευρῶν τετράγωνον ἴσον ᾖ τοῖς ἀπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν τετραγώνοις, ἡ περιεχομένη γωνία ὑπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν ὀρθή ἐστιν"},{"language":"ru","status":"unofficial","text":"I.48, угол против третьей стороны: квадрат третьей равен сумме квадратов первых двух — треугольник прямоугольный"}],"package":"urn:grc:eukleides:clir:pythagoras","strength":"strict"},{"id":"urn:grc:eukleides:clir:pythagoras#TrichotomiaPliris","kind":"constraint","labels":[{"language":"grc","status":"unofficial","text":"πᾶν τρίγωνον ἢ ὀρθογώνιον ἢ ἀμβλυγώνιον ἢ ὀξυγώνιόν ἐστιν"},{"language":"ru","status":"unofficial","text":"инвариант §93.1, ПОЛНОТА: всякий треугольник принадлежит одному из трёх родов — прямоугольному (I.48), тупоугольному (II.12) или остроугольному (II.13); четвёртого рода определение I.21 не знает"}],"package":"urn:grc:eukleides:clir:pythagoras"},{"id":"urn:grc:eukleides:clir:pythagoras#Trigonon","kind":"type_decl","labels":[{"language":"grc","status":"official","text":"τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник — тройка сторон, о которой задан вопрос дела"}],"name":"Trigonon","package":"urn:grc:eukleides:clir:pythagoras"},{"id":"urn:grc:eukleides:clir:pythagoras#TrigononEstin","kind":"rule","labels":[{"language":"grc","status":"official","text":"Παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι"},{"language":"ru","status":"unofficial","text":"I.20: тройка составляет треугольник — каждые две стороны, взятые вместе, строго больше третьей"}],"package":"urn:grc:eukleides:clir:pythagoras","strength":"strict"},{"id":"urn:grc:eukleides:clir:pythagoras#amblygonion","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"ἀμβλυγώνιόν ἐστι τὸ τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник тупоугольный — при одной из трёх позиций квадрат на стороне против угла БОЛЬШЕ суммы квадратов двух других (II.12; определение I.21 — «имеющий тупой угол»)"}],"name":"amblygonion","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#amblygonion/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#orthogonion","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"ὀρθογώνιόν ἐστι τὸ τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник прямоугольный — пифагорово соотношение выполнено при одной из трёх позиций прямого угла (I.48)"}],"name":"orthogonion","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#orthogonion/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#ouk_orthogonion","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"οὐκ ὀρθογώνιόν ἐστι τὸ τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник не прямоугольный — пифагорово соотношение не выполнено ни при одной из трёх позиций угла (контрапозиция I.47)"}],"name":"ouk_orthogonion","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#ouk_orthogonion/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"αἱ τρεῖς πλευραὶ τοῦ τριγώνου"},{"language":"ru","status":"unofficial","text":"три стороны треугольника в порядке записи дела; единица длины у всех трёх одна и та же по построению записи — величины безразмерны (Integer), и вопрос пакета от неё не зависит"}],"name":"pleurai","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/a","labels":[],"name":"a","type":{"name":"urn:law:std#Integer"}},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/b","labels":[],"name":"b","type":{"name":"urn:law:std#Integer"}},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/c","labels":[],"name":"c","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#trigonon_estin","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"τρίγωνόν ἐστιν"},{"language":"ru","status":"unofficial","text":"тройка сторон составляет треугольник (I.20 выполнено при любом сочетании двух сторон)"}],"name":"trigonon_estin","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#trigonon_estin/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":6,"fragments":["urn:grc:eukleides:clir:pythagoras#STO_I20","urn:grc:eukleides:clir:pythagoras#STO_I48"],"jurisdiction":"none","namespace":"urn:grc:eukleides:clir:pythagoras","package":"grc-euclid-pythagoras","title":"Теорема Пифагора и трихотомия треугольников: Начала Евклида I.20, I.47, I.48, II.12, II.13 и определение I.21 — вне юрисдикции государства — доктрина — EXECUTABLE 3 §33.1"}],"caseHash":"sha256:178a653d98191baf51afcd1748824336b6a64df6d70e82cb6b04ba4e23dd9f42","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции 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τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι.\nἜστω γὰρ τρίγωνον τὸ ΑΒΓ· λέγω, ὅτι τοῦ ΑΒΓ τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι, αἱ μὲν ΒΑ, ΑΓ τῆς ΒΓ, αἱ δὲ ΑΒ, ΒΓ τῆς ΑΓ, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ.\n\nΔιήχθω γὰρ ἡ ΒΑ ἐπὶ τὸ Δ σημεῖον, καὶ κείσθω τῇ ΓΑ ἴση ἡ ΑΔ, καὶ ἐπεζεύχθω ἡ ΔΓ. Ἐπεὶ οὖν ἴση ἐστὶν ἡ ΔΑ τῇ ΑΓ, ἴση ἐστὶ καὶ γωνία ἡ ὑπὸ ΑΔΓ τῇ ὑπὸ ΑΓΔ· μείζων ἄρα ἡ ὑπὸ ΒΓΔ τῆς ὑπὸ ΑΔΓ· καὶ ἐπεὶ τρίγωνόν ἐστι τὸ ΔΓΒ μείζονα ἔχον τὴν ὑπὸ ΒΓΔ γωνίαν τῆς ὑπὸ ΒΔΓ, ὑπὸ δὲ τὴν μείζονα γωνίαν ἡ μείζων πλευρὰ ὑποτείνει, ἡ ΔΒ ἄρα τῆς ΒΓ ἐστι μείζων. ἴση δὲ ἡ ΔΑ τῇ ΑΓ· μείζονες ἄρα αἱ ΒΑ, ΑΓ τῆς ΒΓ· ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ μὲν ΑΒ, ΒΓ τῆς ΓΑ μείζονές εἰσιν, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ.\n\nΠαντὸς ἄρα τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι· ὅπερ ἔδει δεῖξαι."}]},{"contentHash":"sha256:d98b3c123fb5ccec91a8aee56d4e86badd316adbe9d1b2b2f52fe9b10c746fa4","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_I_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_I48","kind":"fragment","locator":"article/48","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:cfb24ebed64a0ddaf8a5d1d7912b26dfc81766df519b7f517dbe0aafa649196e","language":"grc","status":"official","text":"Ἐὰν τριγώνου τὸ ἀπὸ μιᾶς τῶν πλευρῶν τετράγωνον ἴσον ᾖ τοῖς ἀπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν τετραγώνοις, ἡ περιεχομένη γωνία ὑπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν ὀρθή ἐστιν.\n\nΤριγώνου γὰρ τοῦ ΑΒΓ τὸ ἀπὸ μιᾶς τῆς ΒΓ πλευρᾶς τετράγωνον ἴσον ἔστω τοῖς ἀπὸ τῶν ΒΑ, ΑΓ πλευρῶν τετραγώνοις· λέγω, ὅτι ὀρθή ἐστιν ἡ ὑπὸ ΒΑΓ γωνία.\n\nἬχθω γὰρ ἀπὸ τοῦ Α σημείου τῇ ΑΓ εὐθείᾳ πρὸς ὀρθὰς ἡ ΑΔ καὶ κείσθω τῇ ΒΑ ἴση ἡ ΑΔ, καὶ ἐπεζεύχθω ἡ ΔΓ. ἐπεὶ ἴση ἐστὶν ἡ ΔΑ τῇ ΑΒ, ἴσον ἐστὶ καὶ τὸ ἀπὸ τῆς ΔΑ τετράγωνον τῷ ἀπὸ τῆς ΑΒ τετραγώνῳ. κοινὸν προσκείσθω τὸ ἀπὸ τῆς ΑΓ τετράγωνον· τὰ ἄρα ἀπὸ τῶν ΔΑ, ΑΓ τετράγωνα ἴσα ἐστὶ τοῖς ἀπὸ τῶν ΒΑ, ΑΓ τετραγώνοις. ἀλλὰ τοῖς μὲν ἀπὸ τῶν ΔΑ, ΑΓ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΔΓ· ὀρθὴ γάρ ἐστιν ἡ ὑπὸ ΔΑΓ γωνία· τοῖς δὲ ἀπὸ τῶν ΒΑ, ΑΓ ἴσον ἐστὶ τὸ ἀπὸ ΒΓ· ὑπόκειται γάρ· τὸ ἄρα ἀπὸ τῆς ΔΓ τετράγωνον ἴσον ἐστὶ τῷ ἀπὸ τῆς ΒΓ τετραγώνῳ· ὥστε καὶ πλευρὰ ἡ ΔΓ τῇ ΒΓ ἐστιν ἴση· καὶ ἐπεὶ ἴση ἐστὶν ἡ ΔΑ τῇ ΑΒ, κοινὴ δὲ ἡ ΑΓ, δύο δὴ αἱ ΔΑ, ΑΓ δύο ταῖς ΒΑ, ΑΓ ἴσαι εἰσίν· καὶ βάσις ἡ ΔΓ βάσει τῇ ΒΓ ἴση· γωνία ἄρα ἡ ὑπὸ ΔΑΓ γωνίᾳ τῇ ὑπὸ ΒΑΓ [ἐστιν] ἴση. ὀρθὴ δὲ ἡ ὑπὸ ΔΑΓ· ὀρθὴ ἄρα καὶ ἡ ὑπὸ ΒΑΓ.\n\nἘὰν ἄρα τριγώνου τὸ ἀπὸ μιᾶς τῶν πλευρῶν τετράγωνον ἴσον ᾖ τοῖς ἀπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν τετραγώνοις, ἡ περιεχομένη γωνία ὑπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν ὀρθή ἐστιν· ὅπερ ἔδει δεῖξαι."}]}],"text":"orthogonion: **TRUE_ONLY** — установлено\nВыведено правом: trigonon_estin(Треугольник); orthogonion(Треугольник)\nПрименены правила: OrthogonionProsTritei, TrigononEstin\nПраво (вне юрисдикции государства): Теорема Пифагора и трихотомия треугольников: Начала Евклида I.20, I.47, I.48, II.12, II.13 и определение I.21 — доктрина — EXECUTABLE 3 §33.1 (programHash sha256:65b5f45c073f…)\nproof-граф: 7 узлов — поле evaluation готово для law_explain"},{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"установлено","missingInputs":[],"truthStatus":"TRUE_ONLY"},"closedEditionRules":[],"derived":["trigonon_estin(Треугольник)","ouk_orthogonion(Треугольник)","oxygonion(Треугольник)"],"derivedOmitted":0,"evaluation":{"proofGraph":{"nodes":[{"attributes":{"assertion":"urn:mcp:case#fact-1"},"conclusion":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":4},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":5},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":6}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#pleurai"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#trigonon_estin"},"evidence":[],"id":"urn:proof:apply:TrigononEstin:9e4adfca921cceb8acd60192e09d8bdac2ab188ea3b749c2627c98c1a18e3530","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:grc:eukleides:clir:pythagoras#TrigononEstin","sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},"v1":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":4},"v2":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":5},"v3":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":6}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#ouk_orthogonion"},"evidence":[],"id":"urn:proof:apply:OukOrthogonionKataPythagoran:1b9c663a5ec76855d199060a0dce76d63e9c3e33c57e9b30f8f5fea624c8b111","kind":"rule_application","premises":["urn:proof:apply:TrigononEstin:9e4adfca921cceb8acd60192e09d8bdac2ab188ea3b749c2627c98c1a18e3530","urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:grc:eukleides:clir:pythagoras#OukOrthogonionKataPythagoran","sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},"v1":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":4},"v2":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":5},"v3":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":6}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#oxygonion"},"evidence":[],"id":"urn:proof:apply:OxygonionKataPasasTasGonias:ebddb9ba838c8cf4a57c726d2a3becf3785fd095321c9dee0b596ced2936f5b3","kind":"rule_application","premises":["urn:proof:apply:TrigononEstin:9e4adfca921cceb8acd60192e09d8bdac2ab188ea3b749c2627c98c1a18e3530","urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:grc:eukleides:clir:pythagoras#OxygonionKataPasasTasGonias","sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},"v1":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":4},"v2":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":5},"v3":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":6}}},{"attributes":{},"conclusion":{"constraint":"urn:grc:eukleides:clir:pythagoras#OxygonionKaiAmblygonionAsymbata","requirementStatus":"NEITHER","status":"UNDETERMINED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:OxygonionKaiAmblygonionAsymbata:aaab86a3277da7a7b32e9c9448686e2f19abfd036e65b3d3aab15444eff7a21d","kind":"constraint_check","premises":["urn:proof:apply:OxygonionKataPasasTasGonias:ebddb9ba838c8cf4a57c726d2a3becf3785fd095321c9dee0b596ced2936f5b3"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:grc:eukleides:clir:pythagoras#OxygonionKaiOrthogonionAsymbata","requirementStatus":"NEITHER","status":"UNDETERMINED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:OxygonionKaiOrthogonionAsymbata:4b9c290ca91a346f360e9a609920aa3640dee8a56789f7692ad622433c96596b","kind":"constraint_check","premises":["urn:proof:apply:OxygonionKataPasasTasGonias:ebddb9ba838c8cf4a57c726d2a3becf3785fd095321c9dee0b596ced2936f5b3"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:grc:eukleides:clir:pythagoras#TrichotomiaPliris","requirementStatus":"TRUE_ONLY","status":"SATISFIED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:TrichotomiaPliris:569c8e53259cbbf84a49a4f0a3c1b3fc434ad2643f8b972ea69a90491f32bdae","kind":"constraint_check","premises":["urn:proof:apply:OxygonionKataPasasTasGonias:ebddb9ba838c8cf4a57c726d2a3becf3785fd095321c9dee0b596ced2936f5b3","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τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν πλευρῶν τετραγώνοις"},{"language":"ru","status":"unofficial","text":"контрапозиция I.47: ни при одной из трёх позиций угла квадрат на стороне не равен сумме квадратов на двух других — треугольник не прямоугольный"}],"package":"urn:grc:eukleides:clir:pythagoras","strength":"strict"},{"id":"urn:grc:eukleides:clir:pythagoras#OxygonionKaiAmblygonionAsymbata","kind":"constraint","labels":[{"language":"grc","status":"unofficial","text":"ἀδύνατον τὸ αὐτὸ τρίγωνον ὀξυγώνιόν τε εἶναι καὶ ἀμβλυγώνιον"},{"language":"ru","status":"unofficial","text":"инвариант §93.1: остроугольный и тупоугольный — разные роды (определение I.21), и один треугольник не может быть обоими"}],"package":"urn:grc:eukleides:clir:pythagoras"},{"id":"urn:grc:eukleides:clir:pythagoras#OxygonionKaiOrthogonionAsymbata","kind":"constraint","labels":[{"language":"grc","status":"unofficial","text":"ἀδύνατον τὸ αὐτὸ τρίγωνον ὀξυγώνιόν τε εἶναι καὶ ὀρθογώνιον"},{"language":"ru","status":"unofficial","text":"инвариант §93.1: остроугольный и прямоугольный — разные роды (определение I.21), и один треугольник не может быть обоими"}],"package":"urn:grc:eukleides:clir:pythagoras"},{"id":"urn:grc:eukleides:clir:pythagoras#OxygonionKataPasasTasGonias","kind":"rule","labels":[{"language":"grc","status":"official","text":"Ἐν τοῖς ὀξυγωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀξεῖαν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἔλαττόν ἐστι τῶν ἀπὸ τῶν τὴν ὀξεῖαν γωνίαν περιεχουσῶν πλευρῶν τετραγώνων"},{"language":"ru","status":"unofficial","text":"II.13 при всех трёх позициях: квадрат на стороне против угла МЕНЬШЕ суммы квадратов двух других при КАЖДОЙ позиции — треугольник остроугольный (определение I.21 требует все три угла острыми)"}],"package":"urn:grc:eukleides:clir:pythagoras","strength":"strict"},{"id":"urn:grc:eukleides:clir:pythagoras#TrichotomiaPliris","kind":"constraint","labels":[{"language":"grc","status":"unofficial","text":"πᾶν τρίγωνον ἢ ὀρθογώνιον ἢ ἀμβλυγώνιον ἢ ὀξυγώνιόν ἐστιν"},{"language":"ru","status":"unofficial","text":"инвариант §93.1, ПОЛНОТА: всякий треугольник принадлежит одному из трёх родов — прямоугольному (I.48), тупоугольному (II.12) или остроугольному (II.13); четвёртого рода определение I.21 не знает"}],"package":"urn:grc:eukleides:clir:pythagoras"},{"id":"urn:grc:eukleides:clir:pythagoras#Trigonon","kind":"type_decl","labels":[{"language":"grc","status":"official","text":"τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник — тройка сторон, о которой задан вопрос дела"}],"name":"Trigonon","package":"urn:grc:eukleides:clir:pythagoras"},{"id":"urn:grc:eukleides:clir:pythagoras#TrigononEstin","kind":"rule","labels":[{"language":"grc","status":"official","text":"Παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι"},{"language":"ru","status":"unofficial","text":"I.20: тройка составляет треугольник — каждые две стороны, взятые вместе, строго больше третьей"}],"package":"urn:grc:eukleides:clir:pythagoras","strength":"strict"},{"id":"urn:grc:eukleides:clir:pythagoras#amblygonion","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"ἀμβλυγώνιόν ἐστι τὸ τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник тупоугольный — при одной из трёх позиций квадрат на стороне против угла БОЛЬШЕ суммы квадратов двух других (II.12; определение I.21 — «имеющий тупой угол»)"}],"name":"amblygonion","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#amblygonion/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#orthogonion","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"ὀρθογώνιόν ἐστι τὸ τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник прямоугольный — пифагорово соотношение выполнено при одной из трёх позиций прямого угла (I.48)"}],"name":"orthogonion","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#orthogonion/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#ouk_orthogonion","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"οὐκ ὀρθογώνιόν ἐστι τὸ τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник не прямоугольный — пифагорово соотношение не выполнено ни при одной из трёх позиций угла (контрапозиция I.47)"}],"name":"ouk_orthogonion","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#ouk_orthogonion/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#oxygonion","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"ὀξυγώνιόν ἐστι τὸ τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник остроугольный — при КАЖДОЙ из трёх позиций квадрат на стороне против угла МЕНЬШЕ суммы квадратов двух других (II.13; определение I.21 — «имеющий все три угла острыми»)"}],"name":"oxygonion","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#oxygonion/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"αἱ τρεῖς πλευραὶ τοῦ τριγώνου"},{"language":"ru","status":"unofficial","text":"три стороны треугольника в порядке записи дела; единица длины у всех трёх одна и та же по построению записи — величины безразмерны (Integer), и вопрос пакета от неё не зависит"}],"name":"pleurai","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/a","labels":[],"name":"a","type":{"name":"urn:law:std#Integer"}},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/b","labels":[],"name":"b","type":{"name":"urn:law:std#Integer"}},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/c","labels":[],"name":"c","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#trigonon_estin","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"τρίγωνόν ἐστιν"},{"language":"ru","status":"unofficial","text":"тройка сторон составляет треугольник (I.20 выполнено при любом сочетании двух сторон)"}],"name":"trigonon_estin","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#trigonon_estin/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":6,"fragments":["urn:grc:eukleides:clir:pythagoras#STO_DEF21","urn:grc:eukleides:clir:pythagoras#STO_I20","urn:grc:eukleides:clir:pythagoras#STO_I47","urn:grc:eukleides:clir:pythagoras#STO_II13"],"jurisdiction":"none","namespace":"urn:grc:eukleides:clir:pythagoras","package":"grc-euclid-pythagoras","title":"Теорема Пифагора и трихотомия треугольников: Начала Евклида I.20, I.47, I.48, II.12, II.13 и определение I.21 — вне юрисдикции государства — доктрина — EXECUTABLE 3 §33.1"}],"caseHash":"sha256:f0707b6739e500659ab5ebba78d5c29b053719e224b8b069a6b623d2e97bea09","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-08-31","mode":"audit","programHash":"sha256:65b5f45c073ff6ca6fa574d0f67dc377bc1fd63eb5fe6ce9b33a2f5354a825fb","resultHash":"sha256:fab0c965e4bf1f71f309571160da2a5a93a7e7a0adf27dfd5fcee970c236e95f","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["Треугольник"],"facts":[{"args":["Треугольник",4,5,6],"predicate":"pleurai"}],"kind":"truth","legalTime":"2026-08-31","package":"grc-euclid-pythagoras","predicate":"oxygonion","proof":true},"sources":[{"contentHash":"sha256:c09651dde2ba61be9043ded2516036ade6ee7dc3a2fce82d0c7ccf7793541363","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_I_HOROI_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_DEF21","kind":"fragment","locator":"article/21","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:151253387f4d7712d81d3d8468268fb49cc4df171e812650bfff5804ab362f38","language":"grc","status":"official","text":"Ἔτι δὲ τῶν τριπλεύρων σχημάτων ὀρθογώνιον μὲν τρίγωνόν ἐστι τὸ ἔχον ὀρθὴν γωνίαν, ἀμβλυγώνιον δὲ τὸ ἔχον ἀμβλεῖαν γωνίαν, ὀξυγώνιον δὲ τὸ τὰς τρεῖς ὀξείας ἔχον γωνίας."}]},{"contentHash":"sha256:3b7073f82734e52fad822a93b48a432a49e26418cc7d7d8811d70699153377d1","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_I_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_I20","kind":"fragment","locator":"article/20","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:15a16264f0261eba60901b9f2645f75ac3d27b6b5a5f42a065ebc0cd32a49a0d","language":"grc","status":"official","text":"Παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι.\nἜστω γὰρ τρίγωνον τὸ ΑΒΓ· λέγω, ὅτι τοῦ ΑΒΓ τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι, αἱ μὲν ΒΑ, ΑΓ τῆς ΒΓ, αἱ δὲ ΑΒ, ΒΓ τῆς ΑΓ, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ.\n\nΔιήχθω γὰρ ἡ ΒΑ ἐπὶ τὸ Δ σημεῖον, καὶ κείσθω τῇ ΓΑ ἴση ἡ ΑΔ, καὶ ἐπεζεύχθω ἡ ΔΓ. Ἐπεὶ οὖν ἴση ἐστὶν ἡ ΔΑ τῇ ΑΓ, ἴση ἐστὶ καὶ γωνία ἡ ὑπὸ ΑΔΓ τῇ ὑπὸ ΑΓΔ· μείζων ἄρα ἡ ὑπὸ ΒΓΔ τῆς ὑπὸ ΑΔΓ· καὶ ἐπεὶ τρίγωνόν ἐστι τὸ ΔΓΒ μείζονα ἔχον τὴν ὑπὸ ΒΓΔ γωνίαν τῆς ὑπὸ ΒΔΓ, ὑπὸ δὲ τὴν μείζονα γωνίαν ἡ μείζων πλευρὰ ὑποτείνει, ἡ ΔΒ ἄρα τῆς ΒΓ ἐστι μείζων. ἴση δὲ ἡ ΔΑ τῇ ΑΓ· μείζονες ἄρα αἱ ΒΑ, ΑΓ τῆς ΒΓ· ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ μὲν ΑΒ, ΒΓ τῆς ΓΑ μείζονές εἰσιν, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ.\n\nΠαντὸς ἄρα τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι· ὅπερ ἔδει δεῖξαι."}]},{"contentHash":"sha256:7b0d82beab361fefa04a3a4aba45fc3581019d568b3d970e12b045da0922fa09","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_I_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_I47","kind":"fragment","locator":"article/47","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:0c330f431aa9aead44c861e9cd4a1cad130e79e1c0f122963e5850ebea2d70f0","language":"grc","status":"official","text":"Ἐν τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν πλευρῶν τετραγώνοις.\n\nἜστω τρίγωνον ὀρθογώνιον τὸ ΑΒΓ ὀρθὴν ἔχον τὴν ὑπὸ ΒΑΓ γωνίαν· λέγω, ὅτι τὸ ἀπὸ τῆς ΒΓ τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΒΑ, ΑΓ τετραγώνοις.\n\nἈναγεγράφθω γὰρ ἀπὸ μὲν τῆς ΒΓ τετράγωνον τὸ ΒΔΕΓ, ἀπὸ δὲ τῶν ΒΑ, ΑΓ τὰ ΗΒ, ΘΓ, καὶ διὰ τοῦ Α ὁποτέρᾳ τῶν ΒΔ, ΓΕ παράλληλος ἤχθω ἡ ΑΛ· καὶ ἐπεζεύχθωσαν αἱ ΑΔ, ΖΓ. καὶ ἐπεὶ ὀρθή ἐστιν ἑκατέρα τῶν ὑπὸ ΒΑΓ, ΒΑΗ γωνιῶν, πρὸς δή τινι εὐθείᾳ τῇ ΒΑ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Α δύο εὐθεῖαι αἱ ΑΓ, ΑΗ μὴ ἐπὶ τὰ αὐτὰ μέρη κείμεναι τὰς ἐφεξῆς γωνίας δυσὶν ὀρθαῖς ἴσας ποιοῦσιν· ἐπ' εὐθείας ἄρα ἐστὶν ἡ ΓΑ τῇ ΑΗ. διὰ τὰ αὐτὰ δὴ καὶ ἡ ΒΑ τῇ ΑΘ ἐστιν ἐπ' εὐθείας. καὶ ἐπεὶ ἴση ἐστὶν ἡ ὑπὸ ΔΒΓ γωνία τῇ ὑπὸ ΖΒΑ· ὀρθὴ γὰρ ἑκατέρα· κοινὴ προσκείσθω ἡ ὑπὸ ΑΒΓ· ὅλη ἄρα ἡ ὑπὸ ΔΒΑ ὅλῃ τῇ ὑπὸ ΖΒΓ ἐστιν ἴση. καὶ ἐπεὶ ἴση ἐστὶν ἡ μὲν ΔΒ τῇ ΒΓ, ἡ δὲ ΖΒ τῇ ΒΑ, δύο δὴ αἱ ΔΒ, ΒΑ δύο ταῖς ΖΒ, ΒΓ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ γωνία ἡ ὑπὸ ΔΒΑ γωνίᾳ τῇ ὑπὸ ΖΒΓ ἴση· βάσις ἄρα ἡ ΑΔ βάσει τῇ ΖΓ [ἐστιν] ἴση, καὶ τὸ ΑΒΔ τρίγωνον τῷ ΖΒΓ τριγώνῳ ἐστὶν ἴσον· καὶ [ἐστὶ] τοῦ μὲν ΑΒΔ τριγώνου διπλάσιον τὸ ΒΛ παραλληλόγραμμον· βάσιν τε γὰρ τὴν αὐτὴν ἔχουσι τὴν ΒΔ καὶ ἐν ταῖς αὐταῖς εἰσι παραλλήλοις ταῖς ΒΔ, ΑΛ· τοῦ δὲ ΖΒΓ τριγώνου διπλάσιον τὸ ΗΒ τετράγωνον· βάσιν τε γὰρ πάλιν τὴν αὐτὴν ἔχουσι τὴν ΖΒ καὶ ἐν ταῖς αὐταῖς εἰσι παραλλήλοις ταῖς ΖΒ, ΗΓ. [τὰ δὲ τῶν ἴσων διπλάσια ἴσα ἀλλήλοις ἐστίν·] ἴσον ἄρα ἐστὶ καὶ τὸ ΒΛ παραλληλόγραμμον τῷ ΗΒ τετραγώνῳ. ὁμοίως δὴ ἐπιζευγνυμένων τῶν ΑΕ, ΒΚ δειχθήσεται καὶ τὸ ΓΛ παραλληλόγραμμον ἴσον τῷ ΘΓ τετραγώνῳ· ὅλον ἄρα τὸ ΒΔΕΓ τετράγωνον δυσὶ τοῖς ΗΒ, ΘΓ τετραγώνοις ἴσον ἐστίν. καί ἐστι τὸ μὲν ΒΔΕΓ τετράγωνον ἀπὸ τῆς ΒΓ ἀναγραφέν, τὰ δὲ ΗΒ, ΘΓ ἀπὸ τῶν ΒΑ, ΑΓ. τὸ ἄρα ἀπὸ τῆς ΒΓ πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΒΑ, ΑΓ πλευρῶν τετραγώνοις.\n\nἘν ἄρα τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν τὴν ὀρθὴν [γωνίαν] περιεχουσῶν πλευρῶν τετραγώνοις· ὅπερ ἔδει δεῖξαι."}]},{"contentHash":"sha256:8e178ea77ee8987496b71af3c4a9ca5ad4be8b6a85e2d2444b28d3f9e81c3603","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_II_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_II13","kind":"fragment","locator":"article/13","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:335b2ee241d2b44bc2a25846ae71637bfe77186c2bdfa56df836555c4c3c7a46","language":"grc","status":"official","text":"Ἐν τοῖς ὀξυγωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀξεῖαν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἔλαττόν ἐστι τῶν ἀπὸ τῶν τὴν ὀξεῖαν γωνίαν περιεχουσῶν πλευρῶν τετραγώνων τῷ περιεχομένῳ δὶς ὑπό τε μιᾶς τῶν περὶ τὴν ὀξεῖαν γωνίαν, ἐφ' ἣν ἡ κάθετος πίπτει, καὶ τῆς ἀπολαμβανομένης ἐντὸς ὑπὸ τῆς καθέτου πρὸς τῇ ὀξείᾳ γωνίᾳ.\n\nἜστω ὀξυγώνιον τρίγωνον τὸ ΑΒΓ ὀξεῖαν ἔχον τὴν πρὸς τῷ Β γωνίαν, καὶ ἤχθω ἀπὸ τοῦ Α σημείου ἐπὶ τὴν ΒΓ κάθετος ἡ ΑΔ· λέγω, ὅτι τὸ ἀπὸ τῆς ΑΓ τετράγωνον ἔλαττόν ἐστι τῶν ἀπὸ τῶν ΓΒ, ΒΑ τετραγώνων τῷ δὶς ὑπὸ τῶν ΓΒ, ΒΔ περιεχομένῳ ὀρθογωνίῳ.\n\nἘπεὶ γὰρ εὐθεῖα ἡ ΓΒ τέτμηται, ὡς ἔτυχεν, κατὰ τὸ Δ, τὰ ἄρα ἀπὸ τῶν ΓΒ, ΒΔ τετράγωνα ἴσα ἐστὶ τῷ τε δὶς ὑπὸ τῶν ΓΒ, ΒΔ περιεχομένῳ ὀρθογωνίῳ καὶ τῷ ἀπὸ τῆς ΔΓ τετραγώνῳ. κοινὸν προσκείσθω τὸ ἀπὸ τῆς ΔΑ τετράγωνον· τὰ ἄρα ἀπὸ τῶν ΓΒ, ΒΔ, ΔΑ τετράγωνα ἴσα ἐστὶ τῷ τε δὶς ὑπὸ τῶν ΓΒ, ΒΔ περιεχομένῳ ὀρθογωνίῳ καὶ τοῖς ἀπὸ τῶν ΑΔ, ΔΓ τετραγώνοις. ἀλλὰ τοῖς μὲν ἀπὸ τῶν ΒΔ, ΔΑ ἴσον τὸ ἀπὸ τῆς ΑΒ· ὀρθὴ γὰρ ἡ πρὸς τῷ Δ γωνίᾳ· τοῖς δὲ ἀπὸ τῶν ΑΔ, ΔΓ ἴσον τὸ ἀπὸ τῆς ΑΓ· τὰ ἄρα ἀπὸ τῶν ΓΒ, ΒΑ ἴσα ἐστὶ τῷ τε ἀπὸ τῆς ΑΓ καὶ τῷ δὶς ὑπὸ τῶν ΓΒ, ΒΔ· ὥστε μόνον τὸ ἀπὸ τῆς ΑΓ ἔλαττόν ἐστι τῶν ἀπὸ τῶν ΓΒ, ΒΑ τετραγώνων τῷ δὶς ὑπὸ τῶν ΓΒ, ΒΔ περιεχομένῳ ὀρθογωνίῳ.\n\nἘν ἄρα τοῖς ὀξυγωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀξεῖαν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἔλαττόν ἐστι τῶν ἀπὸ τῶν τὴν ὀξεῖαν γωνίαν περιεχουσῶν πλευρῶν τετραγώνων τῷ περιεχομένῳ δὶς ὑπό τε μιᾶς τῶν περὶ τὴν ὀξεῖαν γωνίαν, ἐφ' ἣν ἡ κάθετος πίπτει, καὶ τῆς ἀπολαμβανομένης ἐντὸς ὑπὸ τῆς καθέτου πρὸς τῇ ὀξείᾳ γωνίᾳ· ὅπερ ἔδει δεῖξαι."}]}],"text":"oxygonion: **TRUE_ONLY** — установлено\nВыведено правом: trigonon_estin(Треугольник); ouk_orthogonion(Треугольник); oxygonion(Треугольник)\nПрименены правила: OukOrthogonionKataPythagoran, OxygonionKataPasasTasGonias, TrigononEstin\nПраво (вне юрисдикции государства): Теорема Пифагора и трихотомия треугольников: Начала Евклида I.20, I.47, I.48, II.12, II.13 и определение I.21 — доктрина — EXECUTABLE 3 §33.1 (programHash sha256:65b5f45c073f…)\nproof-граф: 8 узлов — поле evaluation готово для 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ἐστιν"},{"language":"ru","status":"unofficial","text":"тройка сторон составляет треугольник (I.20 выполнено при любом сочетании двух сторон)"}],"name":"trigonon_estin","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#trigonon_estin/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":6,"fragments":["urn:grc:eukleides:clir:pythagoras#STO_DEF21","urn:grc:eukleides:clir:pythagoras#STO_I20","urn:grc:eukleides:clir:pythagoras#STO_I47","urn:grc:eukleides:clir:pythagoras#STO_II12"],"jurisdiction":"none","namespace":"urn:grc:eukleides:clir:pythagoras","package":"grc-euclid-pythagoras","title":"Теорема Пифагора и трихотомия треугольников: Начала Евклида I.20, I.47, I.48, II.12, II.13 и определение I.21 — вне юрисдикции государства — доктрина — EXECUTABLE 3 §33.1"}],"caseHash":"sha256:1aac31dea53c8e47f36f0636236ca7918a4faf4e8c23b85d699ce6e19d4fbab0","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-08-31","mode":"audit","programHash":"sha256:65b5f45c073ff6ca6fa574d0f67dc377bc1fd63eb5fe6ce9b33a2f5354a825fb","resultHash":"sha256:f0ced1834c3305efa56fa0143bece660a4b32a6e26926d3e38d07f732a7aa98f","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["Треугольник"],"facts":[{"args":["Треугольник",2,3,4],"predicate":"pleurai"}],"kind":"truth","legalTime":"2026-08-31","package":"grc-euclid-pythagoras","predicate":"amblygonion","proof":true},"sources":[{"contentHash":"sha256:c09651dde2ba61be9043ded2516036ade6ee7dc3a2fce82d0c7ccf7793541363","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_I_HOROI_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_DEF21","kind":"fragment","locator":"article/21","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:151253387f4d7712d81d3d8468268fb49cc4df171e812650bfff5804ab362f38","language":"grc","status":"official","text":"Ἔτι δὲ τῶν τριπλεύρων σχημάτων ὀρθογώνιον μὲν τρίγωνόν ἐστι τὸ ἔχον ὀρθὴν γωνίαν, ἀμβλυγώνιον δὲ τὸ ἔχον ἀμβλεῖαν γωνίαν, ὀξυγώνιον δὲ τὸ τὰς τρεῖς ὀξείας ἔχον γωνίας."}]},{"contentHash":"sha256:3b7073f82734e52fad822a93b48a432a49e26418cc7d7d8811d70699153377d1","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_I_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_I20","kind":"fragment","locator":"article/20","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:15a16264f0261eba60901b9f2645f75ac3d27b6b5a5f42a065ebc0cd32a49a0d","language":"grc","status":"official","text":"Παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι.\nἜστω γὰρ τρίγωνον τὸ ΑΒΓ· λέγω, ὅτι τοῦ ΑΒΓ τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι, αἱ μὲν ΒΑ, ΑΓ τῆς ΒΓ, αἱ δὲ ΑΒ, ΒΓ τῆς ΑΓ, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ.\n\nΔιήχθω γὰρ ἡ ΒΑ ἐπὶ τὸ Δ σημεῖον, καὶ κείσθω τῇ ΓΑ ἴση ἡ ΑΔ, καὶ ἐπεζεύχθω ἡ ΔΓ. Ἐπεὶ οὖν ἴση ἐστὶν ἡ ΔΑ τῇ ΑΓ, ἴση ἐστὶ καὶ γωνία ἡ ὑπὸ ΑΔΓ τῇ ὑπὸ ΑΓΔ· μείζων ἄρα ἡ ὑπὸ ΒΓΔ τῆς ὑπὸ ΑΔΓ· καὶ ἐπεὶ τρίγωνόν ἐστι τὸ ΔΓΒ μείζονα ἔχον τὴν ὑπὸ ΒΓΔ γωνίαν τῆς ὑπὸ ΒΔΓ, ὑπὸ δὲ τὴν μείζονα γωνίαν ἡ μείζων πλευρὰ ὑποτείνει, ἡ ΔΒ ἄρα τῆς ΒΓ ἐστι μείζων. ἴση δὲ ἡ ΔΑ τῇ ΑΓ· μείζονες ἄρα αἱ ΒΑ, ΑΓ τῆς ΒΓ· ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ μὲν ΑΒ, ΒΓ τῆς ΓΑ μείζονές εἰσιν, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ.\n\nΠαντὸς ἄρα τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι· ὅπερ ἔδει δεῖξαι."}]},{"contentHash":"sha256:7b0d82beab361fefa04a3a4aba45fc3581019d568b3d970e12b045da0922fa09","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_I_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_I47","kind":"fragment","locator":"article/47","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:0c330f431aa9aead44c861e9cd4a1cad130e79e1c0f122963e5850ebea2d70f0","language":"grc","status":"official","text":"Ἐν τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν πλευρῶν τετραγώνοις.\n\nἜστω τρίγωνον ὀρθογώνιον τὸ ΑΒΓ ὀρθὴν ἔχον τὴν ὑπὸ ΒΑΓ γωνίαν· λέγω, ὅτι τὸ ἀπὸ τῆς ΒΓ τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΒΑ, ΑΓ τετραγώνοις.\n\nἈναγεγράφθω γὰρ ἀπὸ μὲν τῆς ΒΓ τετράγωνον τὸ ΒΔΕΓ, ἀπὸ δὲ τῶν ΒΑ, ΑΓ τὰ ΗΒ, ΘΓ, καὶ διὰ τοῦ Α ὁποτέρᾳ τῶν ΒΔ, ΓΕ παράλληλος ἤχθω ἡ ΑΛ· καὶ ἐπεζεύχθωσαν αἱ ΑΔ, ΖΓ. καὶ ἐπεὶ ὀρθή ἐστιν ἑκατέρα τῶν ὑπὸ ΒΑΓ, ΒΑΗ γωνιῶν, πρὸς δή τινι εὐθείᾳ τῇ ΒΑ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Α δύο εὐθεῖαι αἱ ΑΓ, ΑΗ μὴ ἐπὶ τὰ αὐτὰ μέρη κείμεναι τὰς ἐφεξῆς γωνίας δυσὶν ὀρθαῖς ἴσας ποιοῦσιν· ἐπ' εὐθείας ἄρα ἐστὶν ἡ ΓΑ τῇ ΑΗ. διὰ τὰ αὐτὰ δὴ καὶ ἡ ΒΑ τῇ ΑΘ ἐστιν ἐπ' εὐθείας. καὶ ἐπεὶ ἴση ἐστὶν ἡ ὑπὸ ΔΒΓ γωνία τῇ ὑπὸ ΖΒΑ· ὀρθὴ γὰρ ἑκατέρα· κοινὴ προσκείσθω ἡ ὑπὸ ΑΒΓ· ὅλη ἄρα ἡ ὑπὸ ΔΒΑ ὅλῃ τῇ ὑπὸ ΖΒΓ ἐστιν ἴση. καὶ ἐπεὶ ἴση ἐστὶν ἡ μὲν ΔΒ τῇ ΒΓ, ἡ δὲ ΖΒ τῇ ΒΑ, δύο δὴ αἱ ΔΒ, ΒΑ δύο ταῖς ΖΒ, ΒΓ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ γωνία ἡ ὑπὸ ΔΒΑ γωνίᾳ τῇ ὑπὸ ΖΒΓ ἴση· βάσις ἄρα ἡ ΑΔ βάσει τῇ ΖΓ [ἐστιν] ἴση, καὶ τὸ ΑΒΔ τρίγωνον τῷ ΖΒΓ τριγώνῳ ἐστὶν ἴσον· καὶ [ἐστὶ] τοῦ μὲν ΑΒΔ τριγώνου διπλάσιον τὸ ΒΛ παραλληλόγραμμον· βάσιν τε γὰρ τὴν αὐτὴν ἔχουσι τὴν ΒΔ καὶ ἐν ταῖς αὐταῖς εἰσι παραλλήλοις ταῖς ΒΔ, ΑΛ· τοῦ δὲ ΖΒΓ τριγώνου διπλάσιον τὸ ΗΒ τετράγωνον· βάσιν τε γὰρ πάλιν τὴν αὐτὴν ἔχουσι τὴν ΖΒ καὶ ἐν ταῖς αὐταῖς εἰσι παραλλήλοις ταῖς ΖΒ, ΗΓ. [τὰ δὲ τῶν ἴσων διπλάσια ἴσα ἀλλήλοις ἐστίν·] ἴσον ἄρα ἐστὶ καὶ τὸ ΒΛ παραλληλόγραμμον τῷ ΗΒ τετραγώνῳ. ὁμοίως δὴ ἐπιζευγνυμένων τῶν ΑΕ, ΒΚ δειχθήσεται καὶ τὸ ΓΛ παραλληλόγραμμον ἴσον τῷ ΘΓ τετραγώνῳ· ὅλον ἄρα τὸ ΒΔΕΓ τετράγωνον δυσὶ τοῖς ΗΒ, ΘΓ τετραγώνοις ἴσον ἐστίν. καί ἐστι τὸ μὲν ΒΔΕΓ τετράγωνον ἀπὸ τῆς ΒΓ ἀναγραφέν, τὰ δὲ ΗΒ, ΘΓ ἀπὸ τῶν ΒΑ, ΑΓ. τὸ ἄρα ἀπὸ τῆς ΒΓ πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΒΑ, ΑΓ πλευρῶν τετραγώνοις.\n\nἘν ἄρα τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν τὴν ὀρθὴν [γωνίαν] περιεχουσῶν πλευρῶν τετραγώνοις· ὅπερ ἔδει δεῖξαι."}]},{"contentHash":"sha256:cf91d742b5cf5ac4bc868c5728247b207217dad65b40ca626fc78e74e4ca078f","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_II_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_II12","kind":"fragment","locator":"article/12","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:d43fba5d23db2f11718c3b0e0b7cb044b06d49165905dcf1c5c0a8122e08bf40","language":"grc","status":"official","text":"Ἐν τοῖς ἀμβλυγωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ἀμβλεῖαν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον μεῖζόν ἐστι τῶν ἀπὸ τῶν τὴν ἀμβλεῖαν γωνίαν περιεχουσῶν πλευρῶν τετραγώνων τῷ περιεχομένῳ δὶς ὑπό τε μιᾶς τῶν περὶ τὴν ἀμβλεῖαν γωνίαν, ἐφ' ἣν ἡ κάθετος πίπτει, καὶ τῆς ἀπολαμβανομένης ἐκτὸς ὑπὸ τῆς καθέτου πρὸς τῇ ἀμβλείᾳ γωνίᾳ.\n\nἜστω ἀμβλυγώνιον τρίγωνον τὸ ΑΒΓ ἀμβλεῖαν ἔχον τὴν ὑπὸ ΒΑΓ, καὶ ἤχθω ἀπὸ τοῦ Β σημείου ἐπὶ τὴν ΓΑ ἐκβληθεῖσαν κάθετος ἡ ΒΔ. λέγω, ὅτι τὸ ἀπὸ τῆς ΒΓ τετράγωνον μεῖζόν ἐστι τῶν ἀπὸ τῶν ΒΑ, ΑΓ τετραγώνων τῷ δὶς ὑπὸ τῶν ΓΑ, ΑΔ περιεχομένῳ ὀρθογωνίῳ.\n\nἘπεὶ γὰρ εὐθεῖα ἡ ΓΑ τέτμηται, ὡς ἔτυχεν, κατὰ τὸ Α σημεῖον, τὸ ἄρα ἀπὸ τῆς ΔΓ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΓΑ, ΑΔ τετραγώνοις καὶ τῷ δὶς ὑπὸ τῶν ΓΑ, ΑΔ περιεχομένῳ ὀρθογωνίῳ. κοινὸν προσκείσθω τὸ ἀπὸ τῆς ΔΒ· τὰ ἄρα ἀπὸ τῶν ΓΔ, ΔΒ ἴσα ἐστὶ τοῖς τε ἀπὸ τῶν ΓΑ, ΑΔ, ΔΒ τετραγώνοις καὶ τῷ δὶς ὑπὸ τῶν ΓΑ, ΑΔ [περιεχομένῳ ὀρθογωνίῳ]. ἀλλὰ τοῖς μὲν ἀπὸ τῶν ΓΔ, ΔΒ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΓΒ· ὀρθὴ γὰρ ἡ πρὸς τῷ Δ γωνία· τοῖς δὲ ἀπὸ τῶν ΑΔ, ΔΒ ἴσον τὸ ἀπὸ τῆς ΑΒ· τὸ ἄρα ἀπὸ τῆς ΓΒ τετράγωνον ἴσον ἐστὶ τοῖς τε ἀπὸ τῶν ΓΑ, ΑΒ τετραγώνοις καὶ τῷ δὶς ὑπὸ τῶν ΓΑ, ΑΔ περιεχομένῳ ὀρθογωνίῳ· ὥστε τὸ ἀπὸ τῆς ΓΒ τετράγωνον τῶν ἀπὸ τῶν ΓΑ, ΑΒ τετραγώνων μεῖζόν ἐστι τῷ δὶς ὑπὸ τῶν ΓΑ, ΑΔ περιεχομένῳ ὀρθογωνίῳ.\n\nἘν ἄρα τοῖς ἀμβλυγωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ἀμβλεῖαν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον μεῖζόν ἐστι τῶν ἀπὸ τῶν τὴν ἀμβλεῖαν γωνίαν περιεχουσῶν πλευρῶν τετραγώνων τῷ περιεχομένῳ δὶς ὑπό τε μιᾶς τῶν περὶ τὴν ἀμβλεῖαν γωνίαν, ἐφ' ἣν ἡ κάθετος πίπτει, καὶ τῆς ἀπολαμβανομένης ἐκτὸς ὑπὸ τῆς καθέτου πρὸς τῇ ἀμβλείᾳ γωνίᾳ· ὅπερ ἔδει δεῖξαι."}]}],"text":"amblygonion: **TRUE_ONLY** — установлено\nВыведено правом: trigonon_estin(Треугольник); amblygonion(Треугольник); ouk_orthogonion(Треугольник)\nПрименены правила: AmblygonionProsTritei, OukOrthogonionKataPythagoran, TrigononEstin\nПраво (вне юрисдикции государства): Теорема Пифагора и трихотомия треугольников: Начала Евклида I.20, I.47, I.48, II.12, II.13 и определение I.21 — доктрина — EXECUTABLE 3 §33.1 (programHash sha256:65b5f45c073f…)\nproof-граф: 6 узлов — поле evaluation готово для law_explain"},{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"установлено","missingInputs":[],"truthStatus":"TRUE_ONLY"},"closedEditionRules":[],"derived":["trigonon_estin(Треугольник)","ouk_orthogonion(Треугольник)","oxygonion(Треугольник)"],"derivedOmitted":0,"evaluation":{"proofGraph":{"nodes":[{"attributes":{"assertion":"urn:mcp:case#fact-1"},"conclusion":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":1},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#pleurai"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#trigonon_estin"},"evidence":[],"id":"urn:proof:apply:TrigononEstin:79cc5b12be7b61bfe12529cdf1a0e0b49c5a4ae167b29ff6ea1208b7af805f18","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:grc:eukleides:clir:pythagoras#TrigononEstin","sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},"v1":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7},"v2":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":1},"v3":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#ouk_orthogonion"},"evidence":[],"id":"urn:proof:apply:OukOrthogonionKataPythagoran:cc8b1caa93546bd5dd75c54ca905a1c51d55f23e0d8f39857c8790b662544f6e","kind":"rule_application","premises":["urn:proof:apply:TrigononEstin:79cc5b12be7b61bfe12529cdf1a0e0b49c5a4ae167b29ff6ea1208b7af805f18","urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:grc:eukleides:clir:pythagoras#OukOrthogonionKataPythagoran","sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},"v1":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7},"v2":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":1},"v3":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#oxygonion"},"evidence":[],"id":"urn:proof:apply:OxygonionKataPasasTasGonias:936a6e86b28ed0e91a29917e0eb12ac61395892b9eadebee989e0ca92b095730","kind":"rule_application","premises":["urn:proof:apply:TrigononEstin:79cc5b12be7b61bfe12529cdf1a0e0b49c5a4ae167b29ff6ea1208b7af805f18","urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:grc:eukleides:clir:pythagoras#OxygonionKataPasasTasGonias","sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},"v1":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7},"v2":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":1},"v3":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7}}},{"attributes":{},"conclusion":{"constraint":"urn:grc:eukleides:clir:pythagoras#OxygonionKaiAmblygonionAsymbata","requirementStatus":"NEITHER","status":"UNDETERMINED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:OxygonionKaiAmblygonionAsymbata:f2ce6778fa2e3f484108d26d3904519fb8d4e92bc84ca5fce7ff955886fccdee","kind":"constraint_check","premises":["urn:proof:apply:OxygonionKataPasasTasGonias:936a6e86b28ed0e91a29917e0eb12ac61395892b9eadebee989e0ca92b095730"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:grc:eukleides:clir:pythagoras#OxygonionKaiOrthogonionAsymbata","requirementStatus":"NEITHER","status":"UNDETERMINED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:OxygonionKaiOrthogonionAsymbata:c6dc14cc03c05875fd6e4bcccb2bdb274a67b73d2266329ba997c0c6be7a12eb","kind":"constraint_check","premises":["urn:proof:apply:OxygonionKataPasasTasGonias:936a6e86b28ed0e91a29917e0eb12ac61395892b9eadebee989e0ca92b095730"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:grc:eukleides:clir:pythagoras#TrichotomiaPliris","requirementStatus":"TRUE_ONLY","status":"SATISFIED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:TrichotomiaPliris:2825b90e7f15f782fdf5c4946f3b32e5f4da6dc7b1c09d4c2fe9bebb9f93df97","kind":"constraint_check","premises":["urn:proof:apply:OxygonionKataPasasTasGonias:936a6e86b28ed0e91a29917e0eb12ac61395892b9eadebee989e0ca92b095730","urn:proof:apply:TrigononEstin:79cc5b12be7b61bfe12529cdf1a0e0b49c5a4ae167b29ff6ea1208b7af805f18"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"literal":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#ouk_orthogonion"},"truthStatus":"TRUE_ONLY"},"evidence":[],"id":"urn:proof:query:mcp","kind":"query_evaluation","premises":["urn:proof:apply:OukOrthogonionKataPythagoran:cc8b1caa93546bd5dd75c54ca905a1c51d55f23e0d8f39857c8790b662544f6e"],"sourceAnchors":[]}],"proofHash":"sha256:02ba5a320111d0d5b238f2075094c467df881d7398bbb037d7cd444045d27364","roots":["urn:proof:constraint:OxygonionKaiAmblygonionAsymbata:f2ce6778fa2e3f484108d26d3904519fb8d4e92bc84ca5fce7ff955886fccdee","urn:proof:constraint:OxygonionKaiOrthogonionAsymbata:c6dc14cc03c05875fd6e4bcccb2bdb274a67b73d2266329ba997c0c6be7a12eb","urn:proof:constraint:TrichotomiaPliris:2825b90e7f15f782fdf5c4946f3b32e5f4da6dc7b1c09d4c2fe9bebb9f93df97","urn:proof:query:mcp"]},"resultHash":"sha256:9d5e4699113a6571fd53ad51d8afbbdd10b6cb7e4eb5c8ca2e1f1317f2242d2b","schemaVersion":"law.core.evaluation/0.2"},"evaluationStatus":"COMPUTED","issues":[],"judgmentRequests":[],"proofRef":"urn:proof:query:mcp","provenance":{"acts":[{"contributed":true,"fragmentCount":6,"fragments":["urn:grc:eukleides:clir:pythagoras#STO_DEF21","urn:grc:eukleides:clir:pythagoras#STO_I20","urn:grc:eukleides:clir:pythagoras#STO_I47","urn:grc:eukleides:clir:pythagoras#STO_II13"],"jurisdiction":"none","namespace":"urn:grc:eukleides:clir:pythagoras","package":"grc-euclid-pythagoras","title":"Теорема Пифагора и трихотомия треугольников: Начала Евклида I.20, I.47, I.48, II.12, II.13 и определение I.21 — вне юрисдикции государства — доктрина — EXECUTABLE 3 §33.1"}],"caseHash":"sha256:142c3a01f569898f707b74db992fdeddf3fd4f06fddabdd1716693243f3e490c","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции 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которой задан вопрос дела"}],"namespace":"urn:grc:eukleides:clir:pythagoras","package":"grc.eukleides.pythagoras"}},"vocab":{}},"vulnerableTo":[],"whyNot":[]},"execution":{"evaluationBytes":"{\"conflicts\":[],\"issues\":[],\"manifest\":{\"artifactHash\":\"sha256:02c9179fad3fa04683f7eed1356fe5b9b52b05590bfeb58f67981500df723e37\",\"calendarSnapshot\":\"\",\"caseHash\":\"sha256:142c3a01f569898f707b74db992fdeddf3fd4f06fddabdd1716693243f3e490c\",\"decisionTime\":\"2026-08-31T12:00:00+05:00\",\"evidenceSnapshotHash\":\"sha256:0000000000000000000000000000000000000000000000000000000000000000\",\"externalSnapshots\":{},\"id\":\"urn:manifest:oracle-1\",\"interpretations\":[],\"knowledgeTime\":\"2026-08-31T12:00:00+05:00\",\"legalTime\":\"2026-08-31\",\"lockfileHash\":\"sha256:0000000000000000000000000000000000000000000000000000000000000000\",\"mode\":\"audit\",\"policies\":{},\"programHash\":\"sha256:65b5f45c073ff6ca6fa574d0f67dc377bc1fd63eb5fe6ce9b33a2f5354a825fb\",\"resolvedEditions\":{},\"semanticHash\":\"sha256:683de8b865ae3f3d7d41baa0122678e4bf2bda57e6e55cc0f9c56c435bafb733\",\"semantics\":\"law.core/0.2\",\"theoryHash\":\"sha256:65b5f45c073ff6ca6fa574d0f67dc377bc1fd63eb5fe6ce9b33a2f5354a825fb\",\"timezone\":\"Asia/Qyzylorda\"},\"positions\":[],\"proofGraph\":{\"nodes\":[{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-1\"},\"conclusion\":{\"args\":[{\"id\":\"urn:mcp:entity:Треугольник\",\"kind\":\"entity_ref\"},{\"kind\":\"value\",\"type\":{\"name\":\"urn:law:std#Integer\"},\"value\":7},{\"kind\":\"value\",\"type\":{\"name\":\"urn:law:std#Integer\"},\"value\":1},{\"kind\":\"value\",\"type\":{\"name\":\"urn:law:std#Integer\"},\"value\":7}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:grc:eukleides:clir:pythagoras#pleurai\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-1\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:mcp:entity:Треугольник\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:grc:eukleides:clir:pythagoras#trigonon_estin\"},\"evidence\":[],\"id\":\"urn:proof:apply:TrigononEstin:79cc5b12be7b61bfe12529cdf1a0e0b49c5a4ae167b29ff6ea1208b7af805f18\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:grc:eukleides:clir:pythagoras#TrigononEstin\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:mcp:entity:Треугольник\",\"kind\":\"entity_ref\"},\"v1\":{\"kind\":\"value\",\"type\":{\"name\":\"urn:law:std#Integer\"},\"value\":7},\"v2\":{\"kind\":\"value\",\"type\":{\"name\":\"urn:law:std#Integer\"},\"value\":1},\"v3\":{\"kind\":\"value\",\"type\":{\"name\":\"urn:law:std#Integer\"},\"value\":7}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:mcp:entity:Треугольник\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:grc:eukleides:clir:pythagoras#ouk_orthogonion\"},\"evidence\":[],\"id\":\"urn:proof:apply:OukOrthogonionKataPythagoran:cc8b1caa93546bd5dd75c54ca905a1c51d55f23e0d8f39857c8790b662544f6e\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:TrigononEstin:79cc5b12be7b61bfe12529cdf1a0e0b49c5a4ae167b29ff6ea1208b7af805f18\",\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:grc:eukleides:clir:pythagoras#OukOrthogonionKataPythagoran\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:mcp:entity:Треугольник\",\"kind\":\"entity_ref\"},\"v1\":{\"kind\":\"value\",\"type\":{\"name\":\"urn:law:std#Integer\"},\"value\":7},\"v2\":{\"kind\":\"value\",\"type\":{\"name\":\"urn:law:std#Integer\"},\"value\":1},\"v3\":{\"kind\":\"value\",\"type\":{\"name\":\"urn:law:std#Integer\"},\"value\":7}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:mcp:entity:Треугольник\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:grc:eukleides:clir:pythagoras#oxygonion\"},\"evidence\":[],\"id\":\"urn:proof:apply:OxygonionKataPasasTasGonias:936a6e86b28ed0e91a29917e0eb12ac61395892b9eadebee989e0ca92b095730\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:TrigononEstin:79cc5b12be7b61bfe12529cdf1a0e0b49c5a4ae167b29ff6ea1208b7af805f18\",\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:grc:eukleides:clir:pythagoras#OxygonionKataPasasTasGonias\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:mcp:entity:Треугольник\",\"kind\":\"entity_ref\"},\"v1\":{\"kind\":\"value\",\"type\":{\"name\":\"urn:law:std#Integer\"},\"value\":7},\"v2\":{\"kind\":\"value\",\"type\":{\"name\":\"urn:law:std#Integer\"},\"value\":1},\"v3\":{\"kind\":\"value\",\"type\":{\"name\":\"urn:law:std#Integer\"},\"value\":7}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:grc:eukleides:clir:pythagoras#OxygonionKaiAmblygonionAsymbata\",\"requirementStatus\":\"NEITHER\",\"status\":\"UNDETERMINED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:OxygonionKaiAmblygonionAsymbata:f2ce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(§93.2)\",\"id\":\"urn:result:constraint:OxygonionKaiOrthogonionAsymbata:162bd5614e90cc00:input:0\",\"kind\":\"fact\",\"relatedNode\":\"urn:grc:eukleides:clir:pythagoras#orthogonion\"}],\"normativeStatus\":\"UNDETERMINED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:OxygonionKaiOrthogonionAsymbata:c6dc14cc03c05875fd6e4bcccb2bdb274a67b73d2266329ba997c0c6be7a12eb\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:grc:eukleides:clir:pythagoras#OxygonionKaiOrthogonionAsymbata\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"NEITHER\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:TrichotomiaPliris:a83297a97a61af6f\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatus\":\"SATISFIED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:TrichotomiaPliris:2825b90e7f15f782fdf5c4946f3b32e5f4da6dc7b1c09d4c2fe9bebb9f93df97\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:grc:eukleides:clir:pythagoras#TrichotomiaPliris\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"TRUE_ONLY\"}],\"schemaVersion\":\"law.core.evaluation/0.2\"}","evaluationSha256":"sha256:fe9e9e1de1e2d95a26f9e3c20bd9735b422ee3d6638ec974c8f083f4a3b25ed7","request":{"case":{"assertions":[{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-1","kind":"assertion","literal":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":1},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#pleurai"},"origin":"case_input","package":"urn:grc:eukleides:clir:pythagoras"}],"context":{"decisionTime":"2026-08-31T12:00:00+05:00","knowledgeTime":"2026-08-31T12:00:00+05:00","legalTime":"2026-08-31","timezone":"Asia/Qyzylorda"},"options":{"selectedInterpretations":[]}},"ir":{"irSha256":"sha256:3421eda7b8097caf79680b6de5b7152cfb789f300f4c66bb6d434debb6ec48c8","kind":"world_ref","nodeCount":42,"packages":[{"artifactHash":"sha256:02c9179fad3fa04683f7eed1356fe5b9b52b05590bfeb58f67981500df723e37","namespace":"urn:grc:eukleides:clir:pythagoras","package":"grc-euclid-pythagoras","semanticHash":"sha256:3fca1331204e68b6c992854a9757652caff8af48c63d48d1a39a0d17f27b84b4"}],"programHash":"sha256:65b5f45c073ff6ca6fa574d0f67dc377bc1fd63eb5fe6ce9b33a2f5354a825fb"},"query":{"kind":"truth","literal":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#ouk_orthogonion"},"queryId":"urn:query:mcp"},"schemaVersion":"law.core.evaluation-request/0.2","semanticVersion":"0.2"},"requestCanonicalSha256":"sha256:b1f03a3f9ab4a54a0c0533e23b2c90b27039f77321f90bf5f296d65c45bae7b1"},"id":"calc-4","kind":"truth","label":"Стороны 7, 1, 7: не прямоугольный","presentation":{"schemaVersion":"law.answers.presentation/0.1","source":{"requestCanonicalSha256":"sha256:b1f03a3f9ab4a54a0c0533e23b2c90b27039f77321f90bf5f296d65c45bae7b1"},"steps":[{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":1},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#pleurai"},"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#trigonon_estin"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#trigonon_estin"},"id":"urn:proof:apply:TrigononEstin:79cc5b12be7b61bfe12529cdf1a0e0b49c5a4ae167b29ff6ea1208b7af805f18","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:grc:eukleides:clir:pythagoras#TrigononEstin","sourceAnchors":["urn:grc:eukleides:clir:pythagoras#STO_I20"],"strength":"strict","substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},"v1":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7},"v2":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":1},"v3":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}},{"id":"v1","type":{"name":"urn:law:std#Integer"}},{"id":"v2","type":{"name":"urn:law:std#Integer"}},{"id":"v3","type":{"name":"urn:law:std#Integer"}}]},{"conclusion":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#ouk_orthogonion"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#ouk_orthogonion"},"id":"urn:proof:apply:OukOrthogonionKataPythagoran:cc8b1caa93546bd5dd75c54ca905a1c51d55f23e0d8f39857c8790b662544f6e","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:TrigononEstin:79cc5b12be7b61bfe12529cdf1a0e0b49c5a4ae167b29ff6ea1208b7af805f18","urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:grc:eukleides:clir:pythagoras#OukOrthogonionKataPythagoran","sourceAnchors":["urn:grc:eukleides:clir:pythagoras#STO_I47"],"strength":"strict","substitution":{"v0":{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"},"v1":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7},"v2":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":1},"v3":{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}},{"id":"v1","type":{"name":"urn:law:std#Integer"}},{"id":"v2","type":{"name":"urn:law:std#Integer"}},{"id":"v3","type":{"name":"urn:law:std#Integer"}}]},{"conclusion":{"literal":{"args":[{"id":"urn:mcp:entity:Треугольник","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:grc:eukleides:clir:pythagoras#ouk_orthogonion"},"truthStatus":"TRUE_ONLY"},"id":"urn:proof:query:mcp","kind":"query_evaluation","originStatus":"available","premises":["urn:proof:apply:OukOrthogonionKataPythagoran:cc8b1caa93546bd5dd75c54ca905a1c51d55f23e0d8f39857c8790b662544f6e"],"trust":"engine"}],"symbols":[{"id":"urn:grc:eukleides:clir:pythagoras#OukOrthogonionKataPythagoran","kind":"rule","labels":[{"language":"grc","status":"official","text":"Ἐν τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν πλευρῶν τετραγώνοις"},{"language":"ru","status":"unofficial","text":"контрапозиция I.47: ни при одной из трёх позиций угла квадрат на стороне не равен сумме квадратов на двух других — треугольник не прямоугольный"}],"package":"urn:grc:eukleides:clir:pythagoras","strength":"strict"},{"id":"urn:grc:eukleides:clir:pythagoras#OxygonionKaiAmblygonionAsymbata","kind":"constraint","labels":[{"language":"grc","status":"unofficial","text":"ἀδύνατον τὸ αὐτὸ τρίγωνον ὀξυγώνιόν τε εἶναι καὶ ἀμβλυγώνιον"},{"language":"ru","status":"unofficial","text":"инвариант §93.1: остроугольный и тупоугольный — разные роды (определение I.21), и один треугольник не может быть обоими"}],"package":"urn:grc:eukleides:clir:pythagoras"},{"id":"urn:grc:eukleides:clir:pythagoras#OxygonionKaiOrthogonionAsymbata","kind":"constraint","labels":[{"language":"grc","status":"unofficial","text":"ἀδύνατον τὸ αὐτὸ τρίγωνον ὀξυγώνιόν τε εἶναι καὶ ὀρθογώνιον"},{"language":"ru","status":"unofficial","text":"инвариант §93.1: остроугольный и прямоугольный — разные роды (определение I.21), и один треугольник не может быть обоими"}],"package":"urn:grc:eukleides:clir:pythagoras"},{"id":"urn:grc:eukleides:clir:pythagoras#OxygonionKataPasasTasGonias","kind":"rule","labels":[{"language":"grc","status":"official","text":"Ἐν τοῖς ὀξυγωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀξεῖαν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἔλαττόν ἐστι τῶν ἀπὸ τῶν τὴν ὀξεῖαν γωνίαν περιεχουσῶν πλευρῶν τετραγώνων"},{"language":"ru","status":"unofficial","text":"II.13 при всех трёх позициях: квадрат на стороне против угла МЕНЬШЕ суммы квадратов двух других при КАЖДОЙ позиции — треугольник остроугольный (определение I.21 требует все три угла острыми)"}],"package":"urn:grc:eukleides:clir:pythagoras","strength":"strict"},{"id":"urn:grc:eukleides:clir:pythagoras#TrichotomiaPliris","kind":"constraint","labels":[{"language":"grc","status":"unofficial","text":"πᾶν τρίγωνον ἢ ὀρθογώνιον ἢ ἀμβλυγώνιον ἢ ὀξυγώνιόν ἐστιν"},{"language":"ru","status":"unofficial","text":"инвариант §93.1, ПОЛНОТА: всякий треугольник принадлежит одному из трёх родов — прямоугольному (I.48), тупоугольному (II.12) или остроугольному (II.13); четвёртого рода определение I.21 не знает"}],"package":"urn:grc:eukleides:clir:pythagoras"},{"id":"urn:grc:eukleides:clir:pythagoras#Trigonon","kind":"type_decl","labels":[{"language":"grc","status":"official","text":"τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник — тройка сторон, о которой задан вопрос дела"}],"name":"Trigonon","package":"urn:grc:eukleides:clir:pythagoras"},{"id":"urn:grc:eukleides:clir:pythagoras#TrigononEstin","kind":"rule","labels":[{"language":"grc","status":"official","text":"Παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι"},{"language":"ru","status":"unofficial","text":"I.20: тройка составляет треугольник — каждые две стороны, взятые вместе, строго больше третьей"}],"package":"urn:grc:eukleides:clir:pythagoras","strength":"strict"},{"id":"urn:grc:eukleides:clir:pythagoras#amblygonion","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"ἀμβλυγώνιόν ἐστι τὸ τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник тупоугольный — при одной из трёх позиций квадрат на стороне против угла БОЛЬШЕ суммы квадратов двух других (II.12; определение I.21 — «имеющий тупой угол»)"}],"name":"amblygonion","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#amblygonion/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#orthogonion","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"ὀρθογώνιόν ἐστι τὸ τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник прямоугольный — пифагорово соотношение выполнено при одной из трёх позиций прямого угла (I.48)"}],"name":"orthogonion","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#orthogonion/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#ouk_orthogonion","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"οὐκ ὀρθογώνιόν ἐστι τὸ τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник не прямоугольный — пифагорово соотношение не выполнено ни при одной из трёх позиций угла (контрапозиция I.47)"}],"name":"ouk_orthogonion","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#ouk_orthogonion/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#oxygonion","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"ὀξυγώνιόν ἐστι τὸ τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник остроугольный — при КАЖДОЙ из трёх позиций квадрат на стороне против угла МЕНЬШЕ суммы квадратов двух других (II.13; определение I.21 — «имеющий все три угла острыми»)"}],"name":"oxygonion","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#oxygonion/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"αἱ τρεῖς πλευραὶ τοῦ τριγώνου"},{"language":"ru","status":"unofficial","text":"три стороны треугольника в порядке записи дела; единица длины у всех трёх одна и та же по построению записи — величины безразмерны (Integer), и вопрос пакета от неё не зависит"}],"name":"pleurai","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/a","labels":[],"name":"a","type":{"name":"urn:law:std#Integer"}},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/b","labels":[],"name":"b","type":{"name":"urn:law:std#Integer"}},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/c","labels":[],"name":"c","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#trigonon_estin","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"τρίγωνόν ἐστιν"},{"language":"ru","status":"unofficial","text":"тройка сторон составляет треугольник (I.20 выполнено при любом сочетании двух сторон)"}],"name":"trigonon_estin","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#trigonon_estin/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":6,"fragments":["urn:grc:eukleides:clir:pythagoras#STO_DEF21","urn:grc:eukleides:clir:pythagoras#STO_I20","urn:grc:eukleides:clir:pythagoras#STO_I47","urn:grc:eukleides:clir:pythagoras#STO_II13"],"jurisdiction":"none","namespace":"urn:grc:eukleides:clir:pythagoras","package":"grc-euclid-pythagoras","title":"Теорема Пифагора и трихотомия треугольников: Начала Евклида I.20, I.47, I.48, II.12, II.13 и определение I.21 — вне юрисдикции государства — доктрина — EXECUTABLE 3 §33.1"}],"caseHash":"sha256:142c3a01f569898f707b74db992fdeddf3fd4f06fddabdd1716693243f3e490c","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-08-31","mode":"audit","programHash":"sha256:65b5f45c073ff6ca6fa574d0f67dc377bc1fd63eb5fe6ce9b33a2f5354a825fb","resultHash":"sha256:9d5e4699113a6571fd53ad51d8afbbdd10b6cb7e4eb5c8ca2e1f1317f2242d2b","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["Треугольник"],"facts":[{"args":["Треугольник",7,1,7],"predicate":"pleurai"}],"kind":"truth","legalTime":"2026-08-31","package":"grc-euclid-pythagoras","predicate":"ouk_orthogonion","proof":true},"sources":[{"contentHash":"sha256:c09651dde2ba61be9043ded2516036ade6ee7dc3a2fce82d0c7ccf7793541363","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_I_HOROI_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_DEF21","kind":"fragment","locator":"article/21","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:151253387f4d7712d81d3d8468268fb49cc4df171e812650bfff5804ab362f38","language":"grc","status":"official","text":"Ἔτι δὲ τῶν τριπλεύρων σχημάτων ὀρθογώνιον μὲν τρίγωνόν ἐστι τὸ ἔχον ὀρθὴν γωνίαν, ἀμβλυγώνιον δὲ τὸ ἔχον ἀμβλεῖαν γωνίαν, ὀξυγώνιον δὲ τὸ τὰς τρεῖς ὀξείας ἔχον γωνίας."}]},{"contentHash":"sha256:3b7073f82734e52fad822a93b48a432a49e26418cc7d7d8811d70699153377d1","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_I_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_I20","kind":"fragment","locator":"article/20","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:15a16264f0261eba60901b9f2645f75ac3d27b6b5a5f42a065ebc0cd32a49a0d","language":"grc","status":"official","text":"Παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι.\nἜστω γὰρ τρίγωνον τὸ ΑΒΓ· λέγω, ὅτι τοῦ ΑΒΓ τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι, αἱ μὲν ΒΑ, ΑΓ τῆς ΒΓ, αἱ δὲ ΑΒ, ΒΓ τῆς ΑΓ, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ.\n\nΔιήχθω γὰρ ἡ ΒΑ ἐπὶ τὸ Δ σημεῖον, καὶ κείσθω τῇ ΓΑ ἴση ἡ ΑΔ, καὶ ἐπεζεύχθω ἡ ΔΓ. Ἐπεὶ οὖν ἴση ἐστὶν ἡ ΔΑ τῇ ΑΓ, ἴση ἐστὶ καὶ γωνία ἡ ὑπὸ ΑΔΓ τῇ ὑπὸ ΑΓΔ· μείζων ἄρα ἡ ὑπὸ ΒΓΔ τῆς ὑπὸ ΑΔΓ· καὶ ἐπεὶ τρίγωνόν ἐστι τὸ ΔΓΒ μείζονα ἔχον τὴν ὑπὸ ΒΓΔ γωνίαν τῆς ὑπὸ ΒΔΓ, ὑπὸ δὲ τὴν μείζονα γωνίαν ἡ μείζων πλευρὰ ὑποτείνει, ἡ ΔΒ ἄρα τῆς ΒΓ ἐστι μείζων. ἴση δὲ ἡ ΔΑ τῇ ΑΓ· μείζονες ἄρα αἱ ΒΑ, ΑΓ τῆς ΒΓ· ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ μὲν ΑΒ, ΒΓ τῆς ΓΑ μείζονές εἰσιν, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ.\n\nΠαντὸς ἄρα τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι· ὅπερ ἔδει δεῖξαι."}]},{"contentHash":"sha256:7b0d82beab361fefa04a3a4aba45fc3581019d568b3d970e12b045da0922fa09","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_I_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_I47","kind":"fragment","locator":"article/47","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:0c330f431aa9aead44c861e9cd4a1cad130e79e1c0f122963e5850ebea2d70f0","language":"grc","status":"official","text":"Ἐν τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν πλευρῶν τετραγώνοις.\n\nἜστω τρίγωνον ὀρθογώνιον τὸ ΑΒΓ ὀρθὴν ἔχον τὴν ὑπὸ ΒΑΓ γωνίαν· λέγω, ὅτι τὸ ἀπὸ τῆς ΒΓ τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΒΑ, ΑΓ τετραγώνοις.\n\nἈναγεγράφθω γὰρ ἀπὸ μὲν τῆς ΒΓ τετράγωνον τὸ ΒΔΕΓ, ἀπὸ δὲ τῶν ΒΑ, ΑΓ τὰ ΗΒ, ΘΓ, καὶ διὰ τοῦ Α ὁποτέρᾳ τῶν ΒΔ, ΓΕ παράλληλος ἤχθω ἡ ΑΛ· καὶ ἐπεζεύχθωσαν αἱ ΑΔ, ΖΓ. καὶ ἐπεὶ ὀρθή ἐστιν ἑκατέρα τῶν ὑπὸ ΒΑΓ, ΒΑΗ γωνιῶν, πρὸς δή τινι εὐθείᾳ τῇ ΒΑ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Α δύο εὐθεῖαι αἱ ΑΓ, ΑΗ μὴ ἐπὶ τὰ αὐτὰ μέρη κείμεναι τὰς ἐφεξῆς γωνίας δυσὶν ὀρθαῖς ἴσας ποιοῦσιν· ἐπ' εὐθείας ἄρα ἐστὶν ἡ ΓΑ τῇ ΑΗ. διὰ τὰ αὐτὰ δὴ καὶ ἡ ΒΑ τῇ ΑΘ ἐστιν ἐπ' εὐθείας. καὶ ἐπεὶ ἴση ἐστὶν ἡ ὑπὸ ΔΒΓ γωνία τῇ ὑπὸ ΖΒΑ· ὀρθὴ γὰρ ἑκατέρα· κοινὴ προσκείσθω ἡ ὑπὸ ΑΒΓ· ὅλη ἄρα ἡ ὑπὸ ΔΒΑ ὅλῃ τῇ ὑπὸ ΖΒΓ ἐστιν ἴση. καὶ ἐπεὶ ἴση ἐστὶν ἡ μὲν ΔΒ τῇ ΒΓ, ἡ δὲ ΖΒ τῇ ΒΑ, δύο δὴ αἱ ΔΒ, ΒΑ δύο ταῖς ΖΒ, ΒΓ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ γωνία ἡ ὑπὸ ΔΒΑ γωνίᾳ τῇ ὑπὸ ΖΒΓ ἴση· βάσις ἄρα ἡ ΑΔ βάσει τῇ ΖΓ [ἐστιν] ἴση, καὶ τὸ ΑΒΔ τρίγωνον τῷ ΖΒΓ τριγώνῳ ἐστὶν ἴσον· καὶ [ἐστὶ] τοῦ μὲν ΑΒΔ τριγώνου διπλάσιον τὸ ΒΛ παραλληλόγραμμον· βάσιν τε γὰρ τὴν αὐτὴν ἔχουσι τὴν ΒΔ καὶ ἐν ταῖς αὐταῖς εἰσι παραλλήλοις ταῖς ΒΔ, ΑΛ· τοῦ δὲ ΖΒΓ τριγώνου διπλάσιον τὸ ΗΒ τετράγωνον· βάσιν τε γὰρ πάλιν τὴν αὐτὴν ἔχουσι τὴν ΖΒ καὶ ἐν ταῖς αὐταῖς εἰσι παραλλήλοις ταῖς ΖΒ, ΗΓ. [τὰ δὲ τῶν ἴσων διπλάσια ἴσα ἀλλήλοις ἐστίν·] ἴσον ἄρα ἐστὶ καὶ τὸ ΒΛ παραλληλόγραμμον τῷ ΗΒ τετραγώνῳ. ὁμοίως δὴ ἐπιζευγνυμένων τῶν ΑΕ, ΒΚ δειχθήσεται καὶ τὸ ΓΛ παραλληλόγραμμον ἴσον τῷ ΘΓ τετραγώνῳ· ὅλον ἄρα τὸ ΒΔΕΓ τετράγωνον δυσὶ τοῖς ΗΒ, ΘΓ τετραγώνοις ἴσον ἐστίν. καί ἐστι τὸ μὲν ΒΔΕΓ τετράγωνον ἀπὸ τῆς ΒΓ ἀναγραφέν, τὰ δὲ ΗΒ, ΘΓ ἀπὸ τῶν ΒΑ, ΑΓ. τὸ ἄρα ἀπὸ τῆς ΒΓ πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΒΑ, ΑΓ πλευρῶν τετραγώνοις.\n\nἘν ἄρα τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν τὴν ὀρθὴν [γωνίαν] περιεχουσῶν πλευρῶν τετραγώνοις· ὅπερ ἔδει δεῖξαι."}]},{"contentHash":"sha256:8e178ea77ee8987496b71af3c4a9ca5ad4be8b6a85e2d2444b28d3f9e81c3603","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_II_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_II13","kind":"fragment","locator":"article/13","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:335b2ee241d2b44bc2a25846ae71637bfe77186c2bdfa56df836555c4c3c7a46","language":"grc","status":"official","text":"Ἐν τοῖς ὀξυγωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀξεῖαν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἔλαττόν ἐστι τῶν ἀπὸ τῶν τὴν ὀξεῖαν γωνίαν περιεχουσῶν πλευρῶν τετραγώνων τῷ περιεχομένῳ δὶς ὑπό τε μιᾶς τῶν περὶ τὴν ὀξεῖαν γωνίαν, ἐφ' ἣν ἡ κάθετος πίπτει, καὶ τῆς ἀπολαμβανομένης ἐντὸς ὑπὸ τῆς καθέτου πρὸς τῇ ὀξείᾳ γωνίᾳ.\n\nἜστω ὀξυγώνιον τρίγωνον τὸ ΑΒΓ ὀξεῖαν ἔχον τὴν πρὸς τῷ Β γωνίαν, καὶ ἤχθω ἀπὸ τοῦ Α σημείου ἐπὶ τὴν ΒΓ κάθετος ἡ ΑΔ· λέγω, ὅτι τὸ ἀπὸ τῆς ΑΓ τετράγωνον ἔλαττόν ἐστι τῶν ἀπὸ τῶν ΓΒ, ΒΑ τετραγώνων τῷ δὶς ὑπὸ τῶν ΓΒ, ΒΔ περιεχομένῳ ὀρθογωνίῳ.\n\nἘπεὶ γὰρ εὐθεῖα ἡ ΓΒ τέτμηται, ὡς ἔτυχεν, κατὰ τὸ Δ, τὰ ἄρα ἀπὸ τῶν ΓΒ, ΒΔ τετράγωνα ἴσα ἐστὶ τῷ τε δὶς ὑπὸ τῶν ΓΒ, ΒΔ περιεχομένῳ ὀρθογωνίῳ καὶ τῷ ἀπὸ τῆς ΔΓ τετραγώνῳ. κοινὸν προσκείσθω τὸ ἀπὸ τῆς ΔΑ τετράγωνον· τὰ ἄρα ἀπὸ τῶν ΓΒ, ΒΔ, ΔΑ τετράγωνα ἴσα ἐστὶ τῷ τε δὶς ὑπὸ τῶν ΓΒ, ΒΔ περιεχομένῳ ὀρθογωνίῳ καὶ τοῖς ἀπὸ τῶν ΑΔ, ΔΓ τετραγώνοις. ἀλλὰ τοῖς μὲν ἀπὸ τῶν ΒΔ, ΔΑ ἴσον τὸ ἀπὸ τῆς ΑΒ· ὀρθὴ γὰρ ἡ πρὸς τῷ Δ γωνίᾳ· τοῖς δὲ ἀπὸ τῶν ΑΔ, ΔΓ ἴσον τὸ ἀπὸ τῆς ΑΓ· τὰ ἄρα ἀπὸ τῶν ΓΒ, ΒΑ ἴσα ἐστὶ τῷ τε ἀπὸ τῆς ΑΓ καὶ τῷ δὶς ὑπὸ τῶν ΓΒ, ΒΔ· ὥστε μόνον τὸ ἀπὸ τῆς ΑΓ ἔλαττόν ἐστι τῶν ἀπὸ τῶν ΓΒ, ΒΑ τετραγώνων τῷ δὶς ὑπὸ τῶν ΓΒ, ΒΔ περιεχομένῳ ὀρθογωνίῳ.\n\nἘν ἄρα τοῖς ὀξυγωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀξεῖαν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἔλαττόν ἐστι τῶν ἀπὸ τῶν τὴν ὀξεῖαν γωνίαν περιεχουσῶν πλευρῶν τετραγώνων τῷ περιεχομένῳ δὶς ὑπό τε μιᾶς τῶν περὶ τὴν ὀξεῖαν γωνίαν, ἐφ' ἣν ἡ κάθετος πίπτει, καὶ τῆς ἀπολαμβανομένης ἐντὸς ὑπὸ τῆς καθέτου πρὸς τῇ ὀξείᾳ γωνίᾳ· ὅπερ ἔδει δεῖξαι."}]}],"text":"ouk_orthogonion: **TRUE_ONLY** — установлено\nВыведено правом: trigonon_estin(Треугольник); ouk_orthogonion(Треугольник); oxygonion(Треугольник)\nПрименены правила: OukOrthogonionKataPythagoran, OxygonionKataPasasTasGonias, TrigononEstin\nПраво (вне юрисдикции государства): Теорема Пифагора и трихотомия треугольников: Начала Евклида I.20, I.47, I.48, II.12, II.13 и определение I.21 — доктрина — EXECUTABLE 3 §33.1 (programHash sha256:65b5f45c073f…)\nproof-граф: 8 узлов — поле evaluation готово для 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треугольника"}],"package":"urn:grc:eukleides:clir:pythagoras","strength":"strict"},{"id":"urn:grc:eukleides:clir:pythagoras#Trigonon","kind":"type_decl","labels":[{"language":"grc","status":"official","text":"τρίγωνον"},{"language":"ru","status":"unofficial","text":"треугольник — тройка сторон, о которой задан вопрос дела"}],"name":"Trigonon","package":"urn:grc:eukleides:clir:pythagoras"},{"id":"urn:grc:eukleides:clir:pythagoras#oude_trigonon","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"οὐδὲ τρίγωνόν ἐστιν"},{"language":"ru","status":"unofficial","text":"тройка сторон не составляет треугольника — какие-то две стороны не больше третьей (I.20 нарушено; равенство — вырождение — тоже 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τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι.\nἜστω γὰρ τρίγωνον τὸ ΑΒΓ· λέγω, ὅτι τοῦ ΑΒΓ τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι, αἱ μὲν ΒΑ, ΑΓ τῆς ΒΓ, αἱ δὲ ΑΒ, ΒΓ τῆς ΑΓ, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ.\n\nΔιήχθω γὰρ ἡ ΒΑ ἐπὶ τὸ Δ σημεῖον, καὶ κείσθω τῇ ΓΑ ἴση ἡ ΑΔ, καὶ ἐπεζεύχθω ἡ ΔΓ. Ἐπεὶ οὖν ἴση ἐστὶν ἡ ΔΑ τῇ ΑΓ, ἴση ἐστὶ καὶ γωνία ἡ ὑπὸ ΑΔΓ τῇ ὑπὸ ΑΓΔ· μείζων ἄρα ἡ ὑπὸ ΒΓΔ τῆς ὑπὸ ΑΔΓ· καὶ ἐπεὶ τρίγωνόν ἐστι τὸ ΔΓΒ μείζονα ἔχον τὴν ὑπὸ ΒΓΔ γωνίαν τῆς ὑπὸ ΒΔΓ, ὑπὸ δὲ τὴν μείζονα γωνίαν ἡ μείζων πλευρὰ ὑποτείνει, ἡ ΔΒ ἄρα τῆς ΒΓ ἐστι μείζων. ἴση δὲ ἡ ΔΑ τῇ ΑΓ· μείζονες ἄρα αἱ ΒΑ, ΑΓ τῆς ΒΓ· ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ μὲν ΑΒ, ΒΓ τῆς ΓΑ μείζονές εἰσιν, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ.\n\nΠαντὸς ἄρα τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι· ὅπερ ἔδει δεῖξαι."}]}],"text":"oude_trigonon: **TRUE_ONLY** — установлено\nВыведено правом: oude_trigonon(Треугольник)\nПрименены правила: OudeTrigononPleuraiAB, OudeTrigononPleuraiAC, OudeTrigononPleuraiBC\nПраво (вне юрисдикции государства): Теорема Пифагора и трихотомия треугольников: Начала Евклида I.20, I.47, I.48, II.12, II.13 и определение I.21 — доктрина — EXECUTABLE 3 §33.1 (programHash sha256:65b5f45c073f…)\nproof-граф: 5 узлов — поле evaluation готово для law_explain"},{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"НЕ УСТАНОВЛЕНО: в формализованном праве нет ни подтверждения, ни опровержения (открытый мир §69) — это не 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дела"}],"namespace":"urn:grc:eukleides:clir:pythagoras","package":"grc.eukleides.pythagoras"}},"vocab":{}},"vulnerableTo":[],"whyNot":[{"anchors":["urn:grc:eukleides:clir:pythagoras#STO_I48"],"computedHead":false,"label":"λέγω, ὅτι ὀρθή ἐστιν ἡ ὑπὸ ΒΑΓ γωνία","premises":[{"premise":"trigonon_estin(Треугольник)","status":"NEITHER"},{"premise":"pleurai(Треугольник, 0, 3, 3)","status":"TRUE_ONLY"},{"note":"терм-гард: операнды не связаны фактами","premise":"v1 × v1 + v3 × v3 = v2 × v2","status":"DEPENDS"}],"rule":"OrthogonionProsDeuterai"},{"anchors":["urn:grc:eukleides:clir:pythagoras#STO_I48"],"computedHead":false,"label":"τριγώνου γὰρ τοῦ ΑΒΓ τὸ ἀπὸ μιᾶς τῆς ΒΓ πλευρᾶς τετράγωνον ἴσον ἔστω τοῖς ἀπὸ τῶν ΒΑ, ΑΓ πλευρῶν τετραγώνοις","premises":[{"note":"терм-гард: операнды не связаны фактами","premise":"v2 × v2 + v3 × v3 = v1 × v1","status":"DEPENDS"},{"premise":"trigonon_estin(Треугольник)","status":"NEITHER"},{"premise":"pleurai(Треугольник, 0, 3, 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квадратов первых двух — треугольник прямоугольный"}],"package":"urn:grc:eukleides:clir:pythagoras","strength":"strict"},{"id":"urn:grc:eukleides:clir:pythagoras#OudeTrigononPleuraiAB","kind":"rule","labels":[{"language":"grc","status":"official","text":"αἱ μὲν ΒΑ, ΑΓ τῆς ΒΓ"},{"language":"ru","status":"unofficial","text":"I.20 нарушено первой парой: первая и вторая стороны, взятые вместе, не больше третьей — тройка не составляет треугольника"}],"package":"urn:grc:eukleides:clir:pythagoras","strength":"strict"},{"id":"urn:grc:eukleides:clir:pythagoras#OudeTrigononPleuraiAC","kind":"rule","labels":[{"language":"grc","status":"official","text":"αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ"},{"language":"ru","status":"unofficial","text":"I.20 нарушено третьей парой: первая и третья стороны, взятые вместе, не больше второй — тройка не составляет 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πλευραὶ τοῦ τριγώνου"},{"language":"ru","status":"unofficial","text":"три стороны треугольника в порядке записи дела; единица длины у всех трёх одна и та же по построению записи — величины безразмерны (Integer), и вопрос пакета от неё не зависит"}],"name":"pleurai","package":"urn:grc:eukleides:clir:pythagoras","parameters":[{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/t","labels":[],"name":"t","type":{"name":"urn:grc:eukleides:clir:pythagoras#Trigonon"}},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/a","labels":[],"name":"a","type":{"name":"urn:law:std#Integer"}},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/b","labels":[],"name":"b","type":{"name":"urn:law:std#Integer"}},{"id":"urn:grc:eukleides:clir:pythagoras#pleurai/arg/c","labels":[],"name":"c","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:grc:eukleides:clir:pythagoras#trigonon_estin","kind":"symbol_decl","labels":[{"language":"grc","status":"unofficial","text":"τρίγωνόν 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государства","legalTime":"2026-08-31","mode":"audit","programHash":"sha256:65b5f45c073ff6ca6fa574d0f67dc377bc1fd63eb5fe6ce9b33a2f5354a825fb","resultHash":"sha256:0dd9c2e7651f1c6001a351a2b0f1457158f80a5c51f1f9390ad7d61bf5500267","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["Треугольник"],"facts":[{"args":["Треугольник",0,3,3],"predicate":"pleurai"}],"kind":"truth","legalTime":"2026-08-31","package":"grc-euclid-pythagoras","predicate":"orthogonion","proof":true},"sources":[{"contentHash":"sha256:3b7073f82734e52fad822a93b48a432a49e26418cc7d7d8811d70699153377d1","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_I_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_I20","kind":"fragment","locator":"article/20","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:15a16264f0261eba60901b9f2645f75ac3d27b6b5a5f42a065ebc0cd32a49a0d","language":"grc","status":"official","text":"Παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι.\nἜστω γὰρ τρίγωνον τὸ ΑΒΓ· λέγω, ὅτι τοῦ ΑΒΓ τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι, αἱ μὲν ΒΑ, ΑΓ τῆς ΒΓ, αἱ δὲ ΑΒ, ΒΓ τῆς ΑΓ, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ.\n\nΔιήχθω γὰρ ἡ ΒΑ ἐπὶ τὸ Δ σημεῖον, καὶ κείσθω τῇ ΓΑ ἴση ἡ ΑΔ, καὶ ἐπεζεύχθω ἡ ΔΓ. Ἐπεὶ οὖν ἴση ἐστὶν ἡ ΔΑ τῇ ΑΓ, ἴση ἐστὶ καὶ γωνία ἡ ὑπὸ ΑΔΓ τῇ ὑπὸ ΑΓΔ· μείζων ἄρα ἡ ὑπὸ ΒΓΔ τῆς ὑπὸ ΑΔΓ· καὶ ἐπεὶ τρίγωνόν ἐστι τὸ ΔΓΒ μείζονα ἔχον τὴν ὑπὸ ΒΓΔ γωνίαν τῆς ὑπὸ ΒΔΓ, ὑπὸ δὲ τὴν μείζονα γωνίαν ἡ μείζων πλευρὰ ὑποτείνει, ἡ ΔΒ ἄρα τῆς ΒΓ ἐστι μείζων. ἴση δὲ ἡ ΔΑ τῇ ΑΓ· μείζονες ἄρα αἱ ΒΑ, ΑΓ τῆς ΒΓ· ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ μὲν ΑΒ, ΒΓ τῆς ΓΑ μείζονές εἰσιν, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ.\n\nΠαντὸς ἄρα τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι· ὅπερ ἔδει δεῖξαι."}]},{"contentHash":"sha256:d98b3c123fb5ccec91a8aee56d4e86badd316adbe9d1b2b2f52fe9b10c746fa4","edition":"urn:grc:eukleides:clir:pythagoras#STOICHEIA_I_GRC","fragmentKind":"article","id":"urn:grc:eukleides:clir:pythagoras#STO_I48","kind":"fragment","locator":"article/48","package":"urn:grc:eukleides:clir:pythagoras","texts":[{"contentHash":"sha256:cfb24ebed64a0ddaf8a5d1d7912b26dfc81766df519b7f517dbe0aafa649196e","language":"grc","status":"official","text":"Ἐὰν τριγώνου τὸ ἀπὸ μιᾶς τῶν πλευρῶν τετράγωνον ἴσον ᾖ τοῖς ἀπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν τετραγώνοις, ἡ περιεχομένη γωνία ὑπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν ὀρθή ἐστιν.\n\nΤριγώνου γὰρ τοῦ ΑΒΓ τὸ ἀπὸ μιᾶς τῆς ΒΓ πλευρᾶς τετράγωνον ἴσον ἔστω τοῖς ἀπὸ τῶν ΒΑ, ΑΓ πλευρῶν τετραγώνοις· λέγω, ὅτι ὀρθή ἐστιν ἡ ὑπὸ ΒΑΓ γωνία.\n\nἬχθω γὰρ ἀπὸ τοῦ Α σημείου τῇ ΑΓ εὐθείᾳ πρὸς ὀρθὰς ἡ ΑΔ καὶ κείσθω τῇ ΒΑ ἴση ἡ ΑΔ, καὶ ἐπεζεύχθω ἡ ΔΓ. ἐπεὶ ἴση ἐστὶν ἡ ΔΑ τῇ ΑΒ, ἴσον ἐστὶ καὶ τὸ ἀπὸ τῆς ΔΑ τετράγωνον τῷ ἀπὸ τῆς ΑΒ τετραγώνῳ. κοινὸν προσκείσθω τὸ ἀπὸ τῆς ΑΓ τετράγωνον· τὰ ἄρα ἀπὸ τῶν ΔΑ, ΑΓ τετράγωνα ἴσα ἐστὶ τοῖς ἀπὸ τῶν ΒΑ, ΑΓ τετραγώνοις. ἀλλὰ τοῖς μὲν ἀπὸ τῶν ΔΑ, ΑΓ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΔΓ· ὀρθὴ γάρ ἐστιν ἡ ὑπὸ ΔΑΓ γωνία· τοῖς δὲ ἀπὸ τῶν ΒΑ, ΑΓ ἴσον ἐστὶ τὸ ἀπὸ ΒΓ· ὑπόκειται γάρ· τὸ ἄρα ἀπὸ τῆς ΔΓ τετράγωνον ἴσον ἐστὶ τῷ ἀπὸ τῆς ΒΓ τετραγώνῳ· ὥστε καὶ πλευρὰ ἡ ΔΓ τῇ ΒΓ ἐστιν ἴση· καὶ ἐπεὶ ἴση ἐστὶν ἡ ΔΑ τῇ ΑΒ, κοινὴ δὲ ἡ ΑΓ, δύο δὴ αἱ ΔΑ, ΑΓ δύο ταῖς ΒΑ, ΑΓ ἴσαι εἰσίν· καὶ βάσις ἡ ΔΓ βάσει τῇ ΒΓ ἴση· γωνία ἄρα ἡ ὑπὸ ΔΑΓ γωνίᾳ τῇ ὑπὸ ΒΑΓ [ἐστιν] ἴση. ὀρθὴ δὲ ἡ ὑπὸ ΔΑΓ· ὀρθὴ ἄρα καὶ ἡ ὑπὸ ΒΑΓ.\n\nἘὰν ἄρα τριγώνου τὸ ἀπὸ μιᾶς τῶν πλευρῶν τετράγωνον ἴσον ᾖ τοῖς ἀπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν τετραγώνοις, ἡ περιεχομένη γωνία ὑπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν ὀρθή ἐστιν· ὅπερ ἔδει δεῖξαι."}]}],"text":"orthogonion: **NEITHER** — НЕ УСТАНОВЛЕНО: в формализованном праве нет ни подтверждения, ни опровержения (открытый мир §69) — это не «нет»\nВыведено правом: oude_trigonon(Треугольник)\nПрименены правила: OudeTrigononPleuraiAB, OudeTrigononPleuraiAC\nПравило «λέγω, ὅτι ὀρθή ἐστιν ἡ ὑπὸ ΒΑΓ γωνία» вывело бы это при посылках:\n    ? trigonon_estin(Треугольник) — NEITHER\n    ✓ pleurai(Треугольник, 0, 3, 3) — TRUE_ONLY\n    · v1 × v1 + v3 × v3 = v2 × v2 — DEPENDS\nПравило «τριγώνου γὰρ τοῦ ΑΒΓ τὸ ἀπὸ μιᾶς τῆς ΒΓ πλευρᾶς τετράγωνον ἴσον ἔστω τοῖς ἀπὸ τῶν ΒΑ, ΑΓ πλευρῶν τετραγώνοις» вывело бы это при посылках:\n    · v2 × v2 + v3 × v3 = v1 × v1 — DEPENDS\n    ? trigonon_estin(Треугольник) — NEITHER\n    ✓ pleurai(Треугольник, 0, 3, 3) — TRUE_ONLY\nПравило «Ἐὰν τριγώνου τὸ ἀπὸ μιᾶς τῶν πλευρῶν τετράγωνον ἴσον ᾖ τοῖς ἀπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν τετραγώνοις, ἡ περιεχομένη γωνία ὑπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν ὀρθή ἐστιν» вывело бы это при посылках:\n    · v1 × v1 + v2 × v2 = v3 × v3 — DEPENDS\n    ? trigonon_estin(Треугольник) — NEITHER\n    ✓ pleurai(Треугольник, 0, 3, 3) — TRUE_ONLY\n(? — факт не подан и не выведен; ✗ — установлено обратное)\nПолные правила с посылками: law_rules({\"predicate\": \"orthogonion\"})\nПраво (вне юрисдикции государства): Теорема Пифагора и трихотомия треугольников: Начала Евклида I.20, I.47, I.48, II.12, II.13 и определение I.21 — доктрина — EXECUTABLE 3 §33.1 (programHash sha256:65b5f45c073f…)\nproof-граф: 4 узлов — поле evaluation готово для law_explain"}],"language":"ru","question":{"origin":"user","text":"Стороны треугольника 3, 4 и 5; 4, 5 и 6; 2, 3 и 4; 0, 0 и 0; 0, 3 и 3. Какой из них прямоугольный по «Началам» Евклида, какой остроугольный, какой тупоугольный, а какой вовсе не треугольник?"},"schemaVersion":"law.answers.document/0.1"}