{"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-similarity"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-similarity-01","title":{"zh-Hant":"相似：以指定正倍率判定標記三角形相似並求面積","en":"Similarity: deciding labeled triangle similarity at a specified positive scale and finding the area"},"summary":{"zh-Hant":"在同一等單位正交平面中，A1=(0,0)、A2=(4,0)、A3=(0,2)，B1=(1,1)、B2=(1,7)、B3=(4,1)，對應 Ai↔Bi，倍率方向為 |BiBj|=k|AiAj|，指定 k=3/2。判定兩者在 k=3/2 下相似，並由已知面積 Area(A)=4 求 Area(B)；再說明 k=2 只被否定為見證、k=−3/2 不合資格，以及方向相反不改變正倍率。","en":"In one equal-unit orthonormal plane, let A1=(0,0), A2=(4,0), A3=(0,2) and B1=(1,1), B2=(1,7), B3=(4,1), with correspondence Ai↔Bi and scale direction |BiBj|=k|AiAj|, at the specified k=3/2. Decide that they are similar at k=3/2, and from the given area Area(A)=4 find Area(B); then explain why k=2 is rejected only as a witness, why k=−3/2 is not eligible, and why opposite orientation does not change the positive scale."},"mko_ids":["mko-similarity"]}],"title":"Concept derivation guides"}