Question 6572: Opposite Midpoint and Three Squared Segments

Original question and givens

In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 22], [-21, 0], [22, 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, 22], [-21, 0], [22, 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

Step-by-step derivation

[[0, 22], [-21, 0], [22, 0]]

original vertex labels1,2,3 in an equal-unit orthogonal Euclidean frame

[[0, 22], [-21, 0], [22, 0]]

three distinct points=Yes

ABC first vector component0 := sub(-21, 0)

ABC first vector component0 = -21

ABC first vector component1 := sub(0, 22)

ABC first vector component1 = -22

ABC second vector component0 := sub(22, 0)

ABC second vector component0 = 22

ABC second vector component1 := sub(0, 22)

ABC second vector component1 = -22

ABC cross positive term := mul(-21, -22)

ABC cross positive term = 462

ABC cross negative term := mul(-22, 22)

ABC cross negative term = -484

ABC signed determinant := sub(462, -484)

ABC signed determinant = 946

distinct=Yes;signed determinant=946

proper triangle=Yes

marked vertex=1

opposite endpoints=[2, 3]

midpoint coordinate sum0 := sum(-21, 22)

midpoint coordinate sum0 = 1

midpoint coordinate0 := div(1, 2)

midpoint coordinate0 = 1/2

midpoint coordinate sum1 := sum(0, 0)

midpoint coordinate sum1 = 0

midpoint coordinate1 := div(0, 2)

midpoint coordinate1 = 0

first endpoint to midpoint coordinate difference0 := sub(-21, 1/2)

first endpoint to midpoint coordinate difference0 = -43/2

first endpoint to midpoint difference squared0 := square(-43/2)

first endpoint to midpoint difference squared0 = 1849/4

first endpoint to midpoint coordinate difference1 := sub(0, 0)

first endpoint to midpoint coordinate difference1 = 0

first endpoint to midpoint difference squared1 := square(0)

first endpoint to midpoint difference squared1 = 0

first endpoint to midpoint squared distance := sum(1849/4, 0)

first endpoint to midpoint squared distance = 1849/4

second endpoint to midpoint coordinate difference0 := sub(22, 1/2)

second endpoint to midpoint coordinate difference0 = 43/2

second endpoint to midpoint difference squared0 := square(43/2)

second endpoint to midpoint difference squared0 = 1849/4

second endpoint to midpoint coordinate difference1 := sub(0, 0)

second endpoint to midpoint coordinate difference1 = 0

second endpoint to midpoint difference squared1 := square(0)

second endpoint to midpoint difference squared1 = 0

second endpoint to midpoint squared distance := sum(1849/4, 0)

second endpoint to midpoint squared distance = 1849/4

marked vertex to midpoint coordinate difference0 := sub(0, 1/2)

marked vertex to midpoint coordinate difference0 = -1/2

marked vertex to midpoint difference squared0 := square(-1/2)

marked vertex to midpoint difference squared0 = 1/4

marked vertex to midpoint coordinate difference1 := sub(22, 0)

marked vertex to midpoint coordinate difference1 = 22

marked vertex to midpoint difference squared1 := square(22)

marked vertex to midpoint difference squared1 = 484

marked vertex to midpoint squared distance := sum(1/4, 484)

marked vertex to midpoint squared distance = 1937/4

endpoint squares=(1849/4,1849/4);vertex-to-midpoint square=1937/4

endpoint equality does not make every median half the opposite side

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, 1849/4, 1849/4, 1937/4)

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.