{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-right-triangle"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-right-triangle-01","title":{"zh-Hant":"直角三角形：用座標判定哪一個內角是直角","en":"Right triangle: deciding in coordinates which interior angle is right"},"summary":{"zh-Hant":"在座標平面模型 ℝ² 中，設 A=(1,1)、B=(5,3)、C=(0,3)。證明 A、B、C 構成非退化三角形，且其在 A 的內角等於 90°（即 π/2），所以 ABC 是直角三角形；說明另外兩角不是直角、任何非退化三角形至多一個直角，並指出斜邊與兩股。再以 D=(1,4) 說明三角形 ABD 不是直角三角形。","en":"In the coordinate-plane model ℝ², let A=(1,1), B=(5,3), C=(0,3). Prove that A, B, C form a nondegenerate triangle whose interior angle at A equals 90° (that is, π/2), so ABC is a right triangle; show that the other two angles are not right and that any nondegenerate triangle has at most one right angle, and identify the hypotenuse and the legs. Then, with D=(1,4), show that triangle ABD is not a right triangle."},"mko_ids":["mko-right-triangle"]}],"title":"概念推導導讀"}