{"schema_version":"ocme-site-results-v1","locale":"en","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-triangle"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-triangle-01","title":{"zh-Hant":"三角形：在座標模型中確認三點界定非退化三角形","en":"Triangle: confirming in the coordinate model that three points determine a nondegenerate triangle"},"summary":{"zh-Hant":"在座標平面模型 ℝ² 中，設 A=(0,0)、B=(4,0)、C=(1,3)。確認 A、B、C 兩兩相異且不共線，因而依 MKO 界定非退化三角形 ABC；寫出三條邊與各頂點的對邊，並說明三頂點的平均點 G=(5/3,1) 不在三角形 ABC 上。另設 D=(2,1)、E=(6,3)，說明相異三點 A、D、E 共線，不界定非退化三角形。","en":"In the coordinate-plane model ℝ², let A=(0,0), B=(4,0), C=(1,3). Confirm that A, B, C are pairwise distinct and noncollinear, so that by the MKO they determine the nondegenerate triangle ABC; write out its three sides and the side opposite each vertex, and show that the average of the vertices, G=(5/3,1), is not on triangle ABC. Then, with D=(2,1) and E=(6,3), show that the distinct points A, D, E are collinear and so do not determine a nondegenerate triangle."},"mko_ids":["mko-triangle"]}],"title":"Concept derivation guides"}