# Question 6563: Opposite Midpoint and Three Squared Segments [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-0088) 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. R0012-06563 · q-20261009-advance-20261013-r07-triangle-0088 ## Original question and givens In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 13], [-12, 0], [13, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers. - G001 — In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 13], [-12, 0], [13, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers. ## 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, 13], [-12, 0], [13, 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, 13], [-12, 0], [13, 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(-12, 0) After ABC first vector component0 = -12 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, 13) After ABC first vector component1 = -13 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(13, 0) After ABC second vector component0 = 13 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, 13) After ABC second vector component1 = -13 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(-12, -13) After ABC cross positive term = 156 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(-13, 13) After ABC cross negative term = -169 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(156, -169) After ABC signed determinant = 325 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=325 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 midpoint coordinate sum0 := sum(-12, 13) After midpoint coordinate sum0 = 1 Conditions: none Add matching coordinates of the opposite endpoints. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore midpoint coordinate0 := div(1, 2) After midpoint coordinate0 = 1/2 Conditions: none Divide the full sum by2,retaining the reduced rational value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore midpoint coordinate sum1 := sum(0, 0) After midpoint coordinate sum1 = 0 Conditions: none Add matching coordinates of the opposite endpoints. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore midpoint coordinate1 := div(0, 2) After midpoint coordinate1 = 0 Conditions: none Divide the full sum by2,retaining the reduced rational value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore first endpoint to midpoint coordinate difference0 := sub(-12, 1/2) After first endpoint to midpoint coordinate difference0 = -25/2 Conditions: none Subtract matching coordinates,allowing rational midpoint coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore first endpoint to midpoint difference squared0 := square(-25/2) After first endpoint to midpoint difference squared0 = 625/4 Conditions: none Square each difference;its sign does not affect squared length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore first endpoint to midpoint coordinate difference1 := sub(0, 0) After first endpoint to midpoint coordinate difference1 = 0 Conditions: none Subtract matching coordinates,allowing rational midpoint coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore first endpoint to midpoint difference squared1 := square(0) After first endpoint to midpoint difference squared1 = 0 Conditions: none Square each difference;its sign does not affect squared length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore first endpoint to midpoint squared distance := sum(625/4, 0) After first endpoint to midpoint squared distance = 625/4 Conditions: none Add the two squares in the equal-unit orthogonal L2 frame. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore second endpoint to midpoint coordinate difference0 := sub(13, 1/2) After second endpoint to midpoint coordinate difference0 = 25/2 Conditions: none Subtract matching coordinates,allowing rational midpoint coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore second endpoint to midpoint difference squared0 := square(25/2) After second endpoint to midpoint difference squared0 = 625/4 Conditions: none Square each difference;its sign does not affect squared length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore second endpoint to midpoint coordinate difference1 := sub(0, 0) After second endpoint to midpoint coordinate difference1 = 0 Conditions: none Subtract matching coordinates,allowing rational midpoint coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore second endpoint to midpoint difference squared1 := square(0) After second endpoint to midpoint difference squared1 = 0 Conditions: none Square each difference;its sign does not affect squared length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore second endpoint to midpoint squared distance := sum(625/4, 0) After second endpoint to midpoint squared distance = 625/4 Conditions: none Add the two squares in the equal-unit orthogonal L2 frame. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore marked vertex to midpoint coordinate difference0 := sub(0, 1/2) After marked vertex to midpoint coordinate difference0 = -1/2 Conditions: none Subtract matching coordinates,allowing rational midpoint coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore marked vertex to midpoint difference squared0 := square(-1/2) After marked vertex to midpoint difference squared0 = 1/4 Conditions: none Square each difference;its sign does not affect squared length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore marked vertex to midpoint coordinate difference1 := sub(13, 0) After marked vertex to midpoint coordinate difference1 = 13 Conditions: none Subtract matching coordinates,allowing rational midpoint coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore marked vertex to midpoint difference squared1 := square(13) After marked vertex to midpoint difference squared1 = 169 Conditions: none Square each difference;its sign does not affect squared length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore marked vertex to midpoint squared distance := sum(1/4, 169) After marked vertex to midpoint squared distance = 677/4 Conditions: none Add the two squares in the equal-unit orthogonal L2 frame. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · impliesBefore endpoint squares=(625/4,625/4);vertex-to-midpoint square=677/4 After endpoint equality does not make every median half the opposite side Conditions: none The midpoint is equally distant from the endpoints,at half the side length.But a general third-vertex median need not have that length;do not apply a right-hypotenuse special case to arbitrary labels. Step identifiers and dependenciesRule: midpoint_endpoint_vs_median_distinction · Uses: G001, S030 ## Reverse checks - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Recompute marked vertex to midpoint squared distance with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute marked vertex to midpoint difference squared1 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute marked vertex to midpoint coordinate difference1 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute marked vertex to midpoint difference squared0 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute marked vertex to midpoint coordinate difference0 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute second endpoint to midpoint squared distance with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute second endpoint to midpoint difference squared1 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute second endpoint to midpoint coordinate difference1 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute second endpoint to midpoint difference squared0 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute second endpoint to midpoint coordinate difference0 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute first endpoint to midpoint squared distance with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute first endpoint to midpoint difference squared1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute first endpoint to midpoint coordinate difference1 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute first endpoint to midpoint difference squared0 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute first endpoint to midpoint coordinate difference0 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute midpoint coordinate1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute midpoint coordinate sum1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute midpoint coordinate0 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute midpoint coordinate sum0 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 (1/2, 0/1, 625/4, 625/4, 677/4) 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: 8db0b38c8ac3f2490dd54372354d2cf04cd48f5d61329e06ce589d40f2856400 - candidate record SHA-256: f9dc8462b437fb7c884f6ef3d5cea1c26357233a91aa3c7b6fda3e72dce5e3f0 - 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-0088.json)