{"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-circle"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-circle-01","title":{"zh-Hant":"圓：以距離平方判定點在圓內、圓上或圓外，並處理零半徑","en":"Circle: classifying points as inside, on or outside by squared distance, including radius zero"},"summary":{"zh-Hant":"在等單位正交座標系中，取中心 O=(2,−1)、半徑 r=5。判定 A=(5,3)、B=(4,1)、E=(−2,3) 各在圓內、圓上或圓外；驗證四個軸向點 O±(5,0)、O±(0,5) 都在 C(O,5) 上，並由兩個相對的軸向點算出直徑長 2r=10。再把半徑改為 r=0，證明 C(O,0)={O}、沒有圓內點，並判定 B 的位置。","en":"In an equal-unit orthonormal frame take center O=(2,−1) and radius r=5. Decide whether each of A=(5,3), B=(4,1), E=(−2,3) is strictly inside, on, or outside the circle; verify that the four axial points O±(5,0), O±(0,5) lie on C(O,5), and compute from two opposite axial points the diameter length 2r=10. Then change the radius to r=0, prove that C(O,0)={O} and that no point is strictly inside, and classify B."},"mko_ids":["mko-circle"]}],"title":"Concept derivation guides"}