# Question 35727: Side Opposite a Vertex [Return to catalogue page 179](https://ocme.evemisslab.com/en/questions/page/179/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260925-triangle-object-language/#q-20260925-triangle-opposite-side-0022) 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. R0070-35727 · q-20260925-triangle-opposite-side-0022 ## Original question and givens Triangle MOQ has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex M? - G001 — Triangle MOQ has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex M? ## 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-triangle](https://ocme.evemisslab.com/en/mko/mko-triangle/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-triangle-opposite-side, [noncollinear, operation, vertex, vertices]] After derived = [triangle, opposite_side, null] Conditions: none Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules. Step identifiers and dependenciesRule: R70_SOURCE_bank_specific_interface · Uses: G001 S002 · impliesBefore source roles = [[M, O, Q], Yes] After derived = [[M, O, Q], Yes, Yes] Conditions: none THIS nondegenerate triangle requires DISTINCT vertices AND GIVEN noncollinearity.Repeated points plus asserted noncollinearity is contradictory.Neither letter names nor a sketch proves noncollinearity. Step identifiers and dependenciesRule: Triangle_DISTINCT_labels_AND_GIVEN_noncollinearity · Uses: G001, S001 S003 · impliesBefore source roles = [M, O] After derived = [MO] Conditions: none EACH unordered pair of distinct vertices defines ONE side.Endpoint sorting is canonical notation only;do not double-count reversed edges or infer lengths. Step identifiers and dependenciesRule: Triangle_EACH_unordered_distinct_endpoint_pair · Uses: G001, S002 S004 · impliesBefore source roles = [M, Q] After derived = [MQ] Conditions: none EACH unordered pair of distinct vertices defines ONE side.Endpoint sorting is canonical notation only;do not double-count reversed edges or infer lengths. Step identifiers and dependenciesRule: Triangle_EACH_unordered_distinct_endpoint_pair · Uses: G001, S003 S005 · impliesBefore source roles = [O, Q] After derived = [OQ] Conditions: none EACH unordered pair of distinct vertices defines ONE side.Endpoint sorting is canonical notation only;do not double-count reversed edges or infer lengths. Step identifiers and dependenciesRule: Triangle_EACH_unordered_distinct_endpoint_pair · Uses: G001, S004 S006 · impliesBefore source roles = [M, [M, O, Q]] After derived = [[O, Q], OQ] Conditions: none The OPPOSITE side joins the TWO vertices remaining after removing the GIVEN vertex.First require that vertex to belong;an incident side is not opposite. Step identifiers and dependenciesRule: Triangle_OPPOSITE_side_excludes_GIVEN_vertex · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S005 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S004 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S003 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S002 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S001 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. ## Result OQ 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: 384a768f3ec2b7b12ed25e085c2d3522a79d89c8d42ed4c55c3ac602679bdfd6 - candidate record SHA-256: 1dfdce0c5ffa700a972fb0a957da825a5baebbc5bf0979136374cacda48ae85a - semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80 - question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260925-triangle-opposite-side-0022.json)