{"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-congruence"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-congruence-01","title":{"zh-Hant":"全等：以 SSS 切片判定有明示對應的三角形全等","en":"Congruence: deciding congruence of labeled triangles by the SSS slice"},"summary":{"zh-Hant":"在同一個等單位正交平面中，A1=(0,0)、A2=(4,0)、A3=(0,3)，B1=(1,1)、B2=(1,5)、B3=(4,1)，對應明示為 Ai↔Bi。先以行列式驗證兩組點都是非退化三角形，再依 12、13、23 的順序精確比較平方邊長，判定 G(A,B)；說明 B 的方向與 A 相反卻仍全等；最後找出另一個對應 A1↔B1、A2↔B3、A3↔B2 的首個不等見證，並說明退化點組不合格。","en":"In one equal-unit orthonormal plane, let A1=(0,0), A2=(4,0), A3=(0,3) and B1=(1,1), B2=(1,5), B3=(4,1), with explicit correspondence Ai↔Bi. First verify by determinants that both triples are nondegenerate triangles, then compare squared side lengths exactly in the order 12, 13, 23 to decide G(A,B); explain why B has the opposite orientation and is still congruent; finally find the first unequal witness for the other correspondence A1↔B1, A2↔B3, A3↔B2, and show that a degenerate triple is not eligible."},"mko_ids":["mko-congruence"]}],"title":"Concept derivation guides"}