Condition
Стороны 3, 4, 5: прямоугольный
Context date 2026-08-31
Calculation result
Established
Input parameters
What we are finding
ὀρθογώνιόν ἐστι τὸ τρίγωνον
Треугольник
Input facts
αἱ τρεῖς πλευραὶ τοῦ τριγώνου
t: Треугольникa: 3b: 4c: 5
Package: 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
Additional details
- Include proof
- Yes
Original data · JSON
{
"args": [
"Треугольник"
],
"facts": [
{
"args": [
"Треугольник",
3,
4,
5
],
"predicate": "pleurai"
}
],
"kind": "truth",
"legalTime": "2026-08-31",
"package": "grc-euclid-pythagoras",
"predicate": "orthogonion",
"proof": true
}Why this resultApplied rules and conditions
Derivation path4 steps
- 1case fact
αἱ τρεῖς πλευραὶ τοῦ τριγώνου
t: Треугольник; a: 3; b: 4; c: 5
- 2rule
Παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι
τρίγωνόν ἐστιν: t: Треугольник
art. 20
Identifier
urn:grc:eukleides:clir:pythagoras#TrigononEstin - 3rule
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
ὀρθογώνιόν ἐστι τὸ τρίγωνον: t: Треугольник
art. 48
Identifier
urn:grc:eukleides:clir:pythagoras#OrthogonionProsTritei - 4query
Query evaluation
verified by the engine: 3 · case fact: 1 · Full graph: 7 nodes
Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.
Basis of this answer
Rules on the saved proof path for this answer.
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
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
Identifier
urn:grc:eukleides:clir:pythagoras#OrthogonionProsTriteiΠαντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι
Identifier
urn:grc:eukleides:clir:pythagoras#TrigononEstin
Derived result for this query
ὀρθογώνιόν ἐστι τὸ τρίγωνον
t: Треугольник
Other derived facts1
τρίγωνόν ἐστιν
t: Треугольник
| t |
|---|
| Треугольник |
| t |
|---|
| Треугольник |
0 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.
Proof graph
Proof nodes: 7 · assertion 1, rule_application 2, constraint_check 3, query_evaluation 1
assertion · urn:proof:assert:urn:mcp:case#fact-1
- attributes
- assertion
- fact-1
- conclusion
- Arguments
- Identifier
- Треугольник
- Type
- entity_ref
- Type
- value
- type
- name
- Integer
- value
- 3
- Type
- value
- type
- name
- Integer
- value
- 4
- Type
- value
- type
- name
- Integer
- value
- 5
- Type
- literal
- Polarity
- positive
- Condition
- pleurai
- evidence
- —
- Identifier
- fact-1
- Type
- assertion
- Premises
- —
- sourceAnchors
- —
rule_application · urn:proof:apply:TrigononEstin:aede226a089a3e3c6e4703def6ebb6193f6cc1a19a686f45cde294caf9255862
- attributes
- —
- conclusion
- Arguments
- Identifier
- Треугольник
- Type
- entity_ref
- Type
- literal
- Polarity
- positive
- Condition
- trigonon_estin
- evidence
- —
- Identifier
- aede226a089a3e3c6e4703def6ebb6193f6cc1a19a686f45cde294caf9255862
- Type
- rule_application
- Premises
- fact-1
- Rule
- TrigononEstin
- sourceAnchors
- —
- substitution
- v0
- Identifier
- Треугольник
- Type
- entity_ref
- v1
- Type
- value
- type
- name
- Integer
- value
- 3
- v2
- Type
- value
- type
- name
- Integer
- value
- 4
- v3
- Type
- value
- type
- name
- Integer
- value
- 5
rule_application · urn:proof:apply:OrthogonionProsTritei:7056bf2dc23ce7e03a854f4450285b51cc7ddbdca773bdba794502331272362a
- attributes
- —
- conclusion
- Arguments
- Identifier
- Треугольник
- Type
- entity_ref
- Type
- literal
- Polarity
- positive
- Condition
- orthogonion
- evidence
- —
- Identifier
- 7056bf2dc23ce7e03a854f4450285b51cc7ddbdca773bdba794502331272362a
- Type
- rule_application
- Premises
- aede226a089a3e3c6e4703def6ebb6193f6cc1a19a686f45cde294caf9255862
- fact-1
- Rule
- OrthogonionProsTritei
- sourceAnchors
- —
- substitution
- v0
- Identifier
- Треугольник
- Type
- entity_ref
- v1
- Type
- value
- type
- name
- Integer
- value
- 3
- v2
- Type
- value
- type
- name
- Integer
- value
- 4
- v3
- Type
- value
- type
- name
- Integer
- value
- 5
constraint_check · urn:proof:constraint:OrthogonionKaiAmblygonionAsymbata:5b3c3e00c9a668a258798c1ee3b3b07267f645534378f22f47fccf14436efa24
- attributes
- —
- conclusion
- constraint
- OrthogonionKaiAmblygonionAsymbata
- requirementStatus
- Not established
- Calculation status
- Undetermined
- triggerStatus
- Satisfied
- evidence
- —
- Identifier
- 5b3c3e00c9a668a258798c1ee3b3b07267f645534378f22f47fccf14436efa24
- Type
- constraint_check
- Premises
- 7056bf2dc23ce7e03a854f4450285b51cc7ddbdca773bdba794502331272362a
- sourceAnchors
- —
- substitution
- v0
- Identifier
- Треугольник
- Type
- entity_ref
constraint_check · urn:proof:constraint:OrthogonionKaiOukOrthogonionAsymbata:6260bd8c1ecf8a49585d04d06414c6c5dc7e0bb851fb49b07666a6f3aba3a047
- attributes
- —
- conclusion
- constraint
- OrthogonionKaiOukOrthogonionAsymbata
- requirementStatus
- Not established
- Calculation status
- Undetermined
- triggerStatus
- Satisfied
- evidence
- —
- Identifier
- 6260bd8c1ecf8a49585d04d06414c6c5dc7e0bb851fb49b07666a6f3aba3a047
- Type
- constraint_check
- Premises
- 7056bf2dc23ce7e03a854f4450285b51cc7ddbdca773bdba794502331272362a
- sourceAnchors
- —
- substitution
- v0
- Identifier
- Треугольник
- Type
- entity_ref
constraint_check · urn:proof:constraint:TrichotomiaPliris:f61efd2e9cb2fa92658918100a35130495ac0310b3993be1c50d4ed98fef2507
- attributes
- —
- conclusion
- constraint
- TrichotomiaPliris
- requirementStatus
- Established
- Calculation status
- Satisfied
- triggerStatus
- Satisfied
- evidence
- —
- Identifier
- f61efd2e9cb2fa92658918100a35130495ac0310b3993be1c50d4ed98fef2507
- Type
- constraint_check
- Premises
- 7056bf2dc23ce7e03a854f4450285b51cc7ddbdca773bdba794502331272362a
- aede226a089a3e3c6e4703def6ebb6193f6cc1a19a686f45cde294caf9255862
- sourceAnchors
- —
- substitution
- v0
- Identifier
- Треугольник
- Type
- entity_ref
query_evaluation · urn:proof:query:mcp
- attributes
- —
- conclusion
- literal
- Arguments
- Identifier
- Треугольник
- Type
- entity_ref
- Type
- literal
- Polarity
- positive
- Condition
- orthogonion
- truthStatus
- Established
- evidence
- —
- Identifier
- mcp
- Type
- query_evaluation
- Premises
- 7056bf2dc23ce7e03a854f4450285b51cc7ddbdca773bdba794502331272362a
- sourceAnchors
- —
Calendar and proof identifiers
- Proof reference
- mcp
Original reasoning · JSON
{
"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": [
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],
"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": [
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"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:6260bd8c1ecf8a49585d04d06414c6c5dc7e0bb851fb49b07666a6f3aba3a047",
"urn:proof:constraint:TrichotomiaPliris:f61efd2e9cb2fa92658918100a35130495ac0310b3993be1c50d4ed98fef2507",
"urn:proof:query:mcp"
]
},
"resultHash": "sha256:1d21e5ee9cd8d7e9e4bb9f4f6c830d182b6e0a3794cf73d251583e48216683fd",
"schemaVersion": "law.core.evaluation/0.2"
},
"proofRef": "urn:proof:query:mcp",
"rulesApplied": [
"urn:grc:eukleides:clir:pythagoras#OrthogonionProsTritei",
"urn:grc:eukleides:clir:pythagoras#TrigononEstin"
]
}SourcesExcerpts: 2
Article 20
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
Παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι. Ἔστω 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. Διήχθω 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. Παντὸς ἄρα τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι· ὅπερ ἔδει δεῖξαι.
Original data · JSON
{
"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."
}
]
}Article 48
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
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. Τριγώνου 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. Ἤχθω 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. Ἐὰν 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.
Original data · JSON
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"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."
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}Packages in the snapshot
- 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
Technical dataFull response, parameters and checksums
- Calculation status
- COMPUTED
Full engine response
orthogonion: TRUE_ONLY — установлено
Выведено правом: trigonon_estin(Треугольник); orthogonion(Треугольник)
Применены правила: OrthogonionProsTritei, TrigononEstin
Право (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…)
proof-граф: 7 узлов — поле evaluation готово для law_explain
Complete machine result · JSON
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"kind": "TRUTH",
"meaning": "установлено",
"missingInputs": [],
"truthStatus": "TRUE_ONLY"
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"closedEditionRules": [],
"derived": [
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"derivedOmitted": 0,
"evaluation": {
"proofGraph": {
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{
"attributes": {
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"conclusion": {
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"kind": "entity_ref"
},
{
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"type": {
"name": "urn:law:std#Integer"
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"value": 3
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{
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"type": {
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"value": 4
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{
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"value": 5
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],
"kind": "literal",
"polarity": "positive",
"predicate": "urn:grc:eukleides:clir:pythagoras#pleurai"
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{
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},
"evidence": [],
"id": "urn:proof:apply:TrigononEstin:aede226a089a3e3c6e4703def6ebb6193f6cc1a19a686f45cde294caf9255862",
"kind": "rule_application",
"premises": [
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"sourceAnchors": [],
"substitution": {
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{
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{
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"triggerStatus": "SATISFIED"
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"evidence": [],
"id": "urn:proof:constraint:OrthogonionKaiOukOrthogonionAsymbata:6260bd8c1ecf8a49585d04d06414c6c5dc7e0bb851fb49b07666a6f3aba3a047",
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{
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"status": "SATISFIED",
"triggerStatus": "SATISFIED"
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"premises": [
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}
},
{
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"conclusion": {
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"kind": "literal",
"polarity": "positive",
"predicate": "urn:grc:eukleides:clir:pythagoras#orthogonion"
},
"truthStatus": "TRUE_ONLY"
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"evidence": [],
"id": "urn:proof:query:mcp",
"kind": "query_evaluation",
"premises": [
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"sourceAnchors": []
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"proofHash": "sha256:58abac9ccca23c56e925f2e137d82a1fea04d448d4982493d24fea9be374ac22",
"roots": [
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"urn:proof:constraint:OrthogonionKaiOukOrthogonionAsymbata:6260bd8c1ecf8a49585d04d06414c6c5dc7e0bb851fb49b07666a6f3aba3a047",
"urn:proof:constraint:TrichotomiaPliris:f61efd2e9cb2fa92658918100a35130495ac0310b3993be1c50d4ed98fef2507",
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},
"resultHash": "sha256:1d21e5ee9cd8d7e9e4bb9f4f6c830d182b6e0a3794cf73d251583e48216683fd",
"schemaVersion": "law.core.evaluation/0.2"
},
"evaluationStatus": "COMPUTED",
"issues": [],
"judgmentRequests": [],
"proofRef": "urn:proof:query:mcp",
"provenance": {
"acts": [
{
"contributed": true,
"fragmentCount": 6,
"fragments": [
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"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 государства",
"legalTime": "2026-08-31",
"mode": "audit",
"programHash": "sha256:65b5f45c073ff6ca6fa574d0f67dc377bc1fd63eb5fe6ce9b33a2f5354a825fb",
"resultHash": "sha256:1d21e5ee9cd8d7e9e4bb9f4f6c830d182b6e0a3794cf73d251583e48216683fd",
"rustCodeHash": "sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4",
"timezone": "Asia/Qyzylorda"
},
"rulesApplied": [
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],
"signature": {
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"parameters": [
{
"labels": [],
"name": "t",
"type": {
"name": "urn:grc:eukleides:clir:pythagoras#Trigonon"
}
}
],
"predicate": "urn:grc:eukleides:clir:pythagoras#orthogonion",
"schemaVersion": "law.answers.signature/0.1",
"types": {
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"kind": "entity",
"labels": [
{
"language": "grc",
"status": "official",
"text": "τρίγωνον"
},
{
"language": "ru",
"status": "unofficial",
"text": "треугольник — тройка сторон, о которой задан вопрос дела"
}
],
"namespace": "urn:grc:eukleides:clir:pythagoras",
"package": "grc.eukleides.pythagoras"
}
},
"vocab": {}
},
"vulnerableTo": [],
"whyNot": []
}Execution · JSON
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},
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},
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"text": "инвариант §93.1: треугольник не может быть одновременно прямоугольным по признаку I.48 и не прямоугольным по контрапозиции I.47"
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"text": "I.48, угол против третьей стороны: квадрат третьей равен сумме квадратов первых двух — треугольник прямоугольный"
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{
"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": [
{
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"type": {
"name": "urn:grc:eukleides:clir:pythagoras#Trigonon"
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}
],
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}
],
"tables": []
}JSON · calculations, sources and exact data
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}- evaluation SHA-256
- sha256:1285a19320357a31546bd027b7bfaa3746a4bd8908ba2e7529464fe3ecd7e089
Original data · JSON
{
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],
"facts": [
{
"args": [
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3,
4,
5
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}