# Question 6597: Labeled Opposite Endpoints and Squared Side [Return to catalogue page 33](https://ocme.evemisslab.com/en/questions/page/33/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261009-advance-20261013-r07-triangle-foundations/#q-20261009-advance-20261013-r07-triangle-0047) 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. R0012-06597 · q-20261009-advance-20261013-r07-triangle-0047 ## Original question and givens In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [22, 0]]. The triangle is qualified as distinct and noncollinear, vertex=1. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance. - G001 — In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [22, 0]]. The triangle is qualified as distinct and noncollinear, vertex=1. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance. ## 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 [[0, 1], [0, 0], [22, 0]] After original vertex labels1,2,3 in an equal-unit orthogonal Euclidean frame Conditions: none Retain the original A,B,C order;the squared-coordinate formula uses equal units and orthogonal axes.Do not read values from a sketch or assume candidate vertices already form a triangle. Step identifiers and dependenciesRule: labelled_triangle_coordinate_scope · Uses: G001 S002 · impliesBefore [[0, 1], [0, 0], [22, 0]] After three distinct points=Yes Conditions: none Check distinctness separately from noncollinearity;collinear points may be distinct,while repeated vertices do not form a proper triangle. Step identifiers and dependenciesRule: distinct_triangle_vertex_check · Uses: G001, S001 S003 · calculationBefore ABC first vector component0 := sub(0, 0) After ABC first vector component0 = 0 Conditions: none The first edge vector is the second vertex minus the first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore ABC first vector component1 := sub(0, 1) After ABC first vector component1 = -1 Conditions: none The first edge vector is the second vertex minus the first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore ABC second vector component0 := sub(22, 0) After ABC second vector component0 = 22 Conditions: none The second edge vector is the third vertex minus that same first vertex. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore ABC second vector component1 := sub(0, 1) After ABC second vector component1 = -1 Conditions: none The second edge vector is the third vertex minus that same first vertex. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore ABC cross positive term := mul(0, -1) After ABC cross positive term = 0 Conditions: none Compute u_x v_y first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore ABC cross negative term := mul(-1, 22) After ABC cross negative term = -22 Conditions: none Compute u_y v_x separately,retaining subtraction for the next step. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore ABC signed determinant := sub(0, -22) After ABC signed determinant = 22 Conditions: none Subtract to obtain signed double area;do not erase label orientation with an absolute value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore distinct=Yes;signed determinant=22 After proper triangle=Yes Conditions: none A proper triangle needs distinct vertices and nonzero determinant;this judges the supplied coordinates,not a general geometric construction proof. Step identifiers and dependenciesRule: triangle_nondegenerate_coordinate_qualification · Uses: G001, S009 S011 · impliesBefore marked vertex=1 After opposite endpoints=[2, 3] Conditions: none The opposite-side endpoints are the other two original labels,not a reordered coordinate or side-length list. Step identifiers and dependenciesRule: original_label_opposite_side · Uses: G001, S010 S012 · calculationBefore opposite side coordinate difference0 := sub(0, 22) After opposite side coordinate difference0 = -22 Conditions: none Subtract matching coordinates,allowing rational midpoint coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore opposite side difference squared0 := square(-22) After opposite side difference squared0 = 484 Conditions: none Square each difference;its sign does not affect squared length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore opposite side coordinate difference1 := sub(0, 0) After opposite side coordinate difference1 = 0 Conditions: none Subtract matching coordinates,allowing rational midpoint coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore opposite side difference squared1 := square(0) After opposite side difference squared1 = 0 Conditions: none Square each difference;its sign does not affect squared length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore opposite side squared distance := sum(484, 0) After opposite side squared distance = 484 Conditions: none Add the two squares in the equal-unit orthogonal L2 frame. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore ([2, 3], 484) After second field is squared length,not length Conditions: none Do not call squared distance the side length or reorder endpoint labels because of the chosen vertex. Step identifiers and dependenciesRule: opposite_side_square_not_length · Uses: G001, S016 ## Reverse checks - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Recompute opposite side squared distance with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute opposite side difference squared1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute opposite side coordinate difference1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute opposite side difference squared0 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute opposite side coordinate difference0 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute ABC signed determinant with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute ABC cross negative term with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute ABC cross positive term with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute ABC second vector component1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute ABC second vector component0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute ABC first vector component1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute ABC first vector component0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([2, 3], 484) 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: 8f7f2da52ceb9ccff1d56aa244442e0fb09b55d0cff9a01f2ed386a899d6d540 - candidate record SHA-256: 6d6c1d82bc5d39d99c3d898b8d7d40ca03e2ef4356d00fcca9f945f5327c486f - semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00 - question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261009-advance-20261013-r07-triangle-0047.json)