{
  "$schema":"../../../schemas/mko-v0.4.schema.json","schema_version":"mko-v0.4","id":"mko-triangle","semantic_family_id":"sf-triangle","type":"concept","version":"0.10.0","language":"zh-Hant",
  "titles":{"zh-Hant":"三角形","en":"Triangle"},
  "summary":{"zh-Hant":"在平面中，三個相異且不共線的點與連接它們的三條線段形成非退化三角形。"},
  "statement":{"zh-Hant":"給定平面中三個相異且不共線的點 A、B、C，線段 AB、BC、CA 的聯合界定非退化三角形 ABC。"},
  "formula":{"tex":"ABC","mathml":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ABC</mi></math>","semantic_ast":{"type":"symbol","name":"ABC","glyph":"ABC"},"compiler":{"id":"ocme-formula-core","version":"0.4.0","source_sha256":"b5d4045c3f466fa91fe2cc6abe79232a1a57cdf104f7a26e716e0a1e2789df78"}},
  "assumptions":[{"id":"a1","zh-Hant":"採平面歐幾里得幾何，且 A、B、C 是相異不共線點。"}],
  "symbols":[{"token":"A","role":"vertex","role_zh":"頂點 A","scope":"triangle_label"},{"token":"B","role":"vertex","role_zh":"頂點 B","scope":"triangle_label"},{"token":"C","role":"vertex","role_zh":"頂點 C","scope":"triangle_label"}],
  "dependencies":[{"id":"mko-point-line-plane","title_zh":"點、線、面","reason_zh":"三角形以平面中的點、不共線條件與連接線段建立。"}],
  "explanation":{"paragraphs_zh":["三角形 ABC 的三個頂點是 A、B、C，三條邊是 AB、BC、CA。頂點順序可改變名稱，但不應憑示意圖外觀增加未明示性質。","三點若有重合或全部共線，就不形成本條目所指的非退化三角形。只說『有三個點』仍不足。","本物件先建立頂點、邊與對邊關係；按邊或角分類、內角和、面積、全等與相似需要額外資料與定理。"],"common_misconception_zh":"常見誤解是任何三點都形成非退化三角形、把頂點名稱當成邊，或從畫圖比例猜邊長與角度。"},
  "proofs":[],"computational_companions":[],
  "verification":{"computational_status":"not_evaluated_no_computational_claim","warning_zh":"明示的點、邊與不共線資料可做角色重算，但不構成三角形存在性、內角和或分類定理的形式證明。","evidence_refs":[],"producers":[{"id":"lean-mathlib","adapter":"formal_system","config_path":"formal/lean/lakefile.toml","status":"configured"}]},
  "formalization":{"status":"not_formalized","target_systems":["Lean 4","Mathlib"],"next_obligation_zh":"選定 EuclideanGeometry 模型並綁定三個相異不共線點、線段與非退化三角形聲明。"},
  "provenance":{"origin":"ocme_local_ai_materialization_candidate","note_zh":"依 research/atlas-triangle.md 整理；待人工與形式化審查。","content_license":"CC-BY-SA-4.0","code_license":"MIT","created_at":"2026-09-25"}
}