# Question 11126: A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It [Return to catalogue page 56](https://ocme.evemisslab.com/en/questions/page/56/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261024-r07-convex-set-foundations/#q-20261013-advance-20261024-r07-convex-set-0076) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0021-11126 · q-20261013-advance-20261024-r07-convex-set-0076 ## Original question and givens Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [2, 0], [0, 2]] retain input order; endpoints left=[2, 0], right=[0, 2] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. - G001 — Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [2, 0], [0, 2]] retain input order; endpoints left=[2, 0], right=[0, 2] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-convex-set](https://ocme.evemisslab.com/en/mko/mko-convex-set/) ## Step-by-step derivation S001 · impliesBefore ordered finite point set=[[0, 0], [2, 0], [0, 2]];left=[2, 0];right=[0, 2] After both endpoints belong;choose lambda=1/2 Conditions: none A counterexample requires both endpoints to be members.Choose the permitted lambda1/2 and compute the exact midpoint,not a visual estimate. Step identifiers and dependenciesRule: midpoint_member_premises · Uses: G001 S002 · calculationBefore midpoint coordinate0 numerator := sum(2, 0) After midpoint coordinate0 numerator = 2 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore midpoint coordinate0 := div(2, 2) After midpoint coordinate0 = 1 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore midpoint coordinate1 numerator := sum(0, 2) After midpoint coordinate1 numerator = 2 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore midpoint coordinate1 := div(2, 2) After midpoint coordinate1 = 1 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore candidate point index=0;point=[0, 0];midpoint=[1, 1] After all coordinates equal=No Conditions: none Compare every set member;all coordinates must match,not merely one. Step identifiers and dependenciesRule: midpoint_complete_set_comparison · Uses: G001, S005 S007 · impliesBefore candidate point index=1;point=[2, 0];midpoint=[1, 1] After all coordinates equal=No Conditions: none Compare every set member;all coordinates must match,not merely one. Step identifiers and dependenciesRule: midpoint_complete_set_comparison · Uses: G001, S006 S008 · impliesBefore candidate point index=2;point=[0, 2];midpoint=[1, 1] After all coordinates equal=No Conditions: none Compare every set member;all coordinates must match,not merely one. Step identifiers and dependenciesRule: midpoint_complete_set_comparison · Uses: G001, S007 S009 · impliesBefore all point comparisons=[No, No, No] After midpoint present=No;this nonconvex witness=Yes Conditions: none An absent midpoint is one exact counterexample refuting convexity.A present midpoint only defeats this counterexample;it does not prove the set convex. Step identifiers and dependenciesRule: midpoint_failed_counterexample_not_convexity · Uses: G001, S008 ## Reverse checks - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute midpoint coordinate1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute midpoint coordinate1 numerator with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute midpoint coordinate0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute midpoint coordinate0 numerator with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([[0, 0], [2, 0], [0, 2]], [2, 0], [0, 2], [1, 1], No, Yes) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 077df9aab245fef266ca44c2a55d670823b5dd94019e1414d747d1ce116128a8 - candidate record SHA-256: 82b75a9933232948562e65de75ac00a8dd2334364a56a4ee5d75a3f5fdc82fe3 - semantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258 - question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261024-r07-convex-set-0076.json)