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R0070-35730 · q-20260925-triangle-opposite-side-0025
Triangle PRT has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex P?
G001 — Triangle PRT has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex P?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.
Before
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.
Rule: R70_SOURCE_bank_specific_interface · Uses: G001
Before
source roles = [[P, R, T], Yes]
After
derived = [[P, R, T], 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.
Rule: Triangle_DISTINCT_labels_AND_GIVEN_noncollinearity · Uses: G001, S001
Before
source roles = [P, R]
After
derived = [PR]
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.
Rule: Triangle_EACH_unordered_distinct_endpoint_pair · Uses: G001, S002
Before
source roles = [P, T]
After
derived = [PT]
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.
Rule: Triangle_EACH_unordered_distinct_endpoint_pair · Uses: G001, S003
Before
source roles = [R, T]
After
derived = [RT]
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.
Rule: Triangle_EACH_unordered_distinct_endpoint_pair · Uses: G001, S004
Before
source roles = [P, [P, R, T]]
After
derived = [[R, T], RT]
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.
Rule: Triangle_OPPOSITE_side_excludes_GIVEN_vertex · Uses: G001, S005
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.RT
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: c136144bb4460c14ecaf9f455a63e5044b89725f9d6a2aabad62de02a9b0f214candidate record SHA-256: 2a86fff81e80edd23f3a8eeb69224b9ccb20b4aebcf7fb8e9627167f1028bbddsemantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dabLean: NOT_RUN · human review: NOT_PERFORMED