Question 30196: Exact Midpoint and Two Squared Segment Lengths

Original question and givens

Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-20, 1], Q=[21, 22], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.

• G001 — Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-20, 1], Q=[21, 22], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). 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-euclidean-length

Step-by-step derivation

source roles = [qf-density12-length-midpoint-equidistance, density12_length_midpoint_equidistance, integer_euclidean_plane]

derived = [[midpoint_x, midpoint_y, SQUARED_PM, SQUARED_MQ]]

Length original PQ ORIGINAL Q minus P coordinate0 := sub(21, -20)

Length original PQ ORIGINAL Q minus P coordinate0 = 41

Length original PQ ORIGINAL Q minus P coordinate1 := sub(22, 1)

Length original PQ ORIGINAL Q minus P coordinate1 = 21

Length original PQ ORIGINAL square coordinate0 := mul(41, 41)

Length original PQ ORIGINAL square coordinate0 = 1681

Length original PQ ORIGINAL square coordinate1 := mul(21, 21)

Length original PQ ORIGINAL square coordinate1 = 441

Length original PQ ORIGINAL squared norm sum := sum(1681, 441)

Length original PQ ORIGINAL squared norm sum = 2122

source roles = [Length original PQ, [-20, 1], [21, 22]]

derived = [[41, 21], 2122]

Midpoint ORIGINAL endpoint sum coordinate0 := sum(-20, 21)

Midpoint ORIGINAL endpoint sum coordinate0 = 1

Midpoint ORIGINAL average coordinate0 := div(1, 2)

Midpoint ORIGINAL average coordinate0 = 1/2

Midpoint ORIGINAL endpoint sum coordinate1 := sum(1, 22)

Midpoint ORIGINAL endpoint sum coordinate1 = 23

Midpoint ORIGINAL average coordinate1 := div(23, 2)

Midpoint ORIGINAL average coordinate1 = 23/2

Midpoint ORIGINAL full squared length over four := div(2122, 4)

Midpoint ORIGINAL full squared length over four = 1061/2

Midpoint reverse PM displacement coordinate0 := sub(1/2, -20)

Midpoint reverse PM displacement coordinate0 = 41/2

Midpoint reverse PM displacement coordinate1 := sub(23/2, 1)

Midpoint reverse PM displacement coordinate1 = 21/2

Midpoint reverse PM coordinate0 square := mul(41/2, 41/2)

Midpoint reverse PM coordinate0 square = 1681/4

Midpoint reverse PM coordinate1 square := mul(21/2, 21/2)

Midpoint reverse PM coordinate1 square = 441/4

Midpoint reverse PM squared sum := sum(1681/4, 441/4)

Midpoint reverse PM squared sum = 1061/2

Midpoint reverse MQ displacement coordinate0 := sub(21, 1/2)

Midpoint reverse MQ displacement coordinate0 = 41/2

Midpoint reverse MQ displacement coordinate1 := sub(22, 23/2)

Midpoint reverse MQ displacement coordinate1 = 21/2

Midpoint reverse MQ coordinate0 square := mul(41/2, 41/2)

Midpoint reverse MQ coordinate0 square = 1681/4

Midpoint reverse MQ coordinate1 square := mul(21/2, 21/2)

Midpoint reverse MQ coordinate1 square = 441/4

Midpoint reverse MQ squared sum := sum(1681/4, 441/4)

Midpoint reverse MQ squared sum = 1061/2

source roles = [[-20, 1], [21, 22], [1/2, 23/2], 2122]

derived = [1061/2, 1061/2, 1061/2]

Reverse checks

• S023 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
• S022 · premise_and_result_audit — Recompute Midpoint reverse MQ squared sum with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Midpoint reverse MQ coordinate1 square with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Midpoint reverse MQ coordinate0 square with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Midpoint reverse MQ displacement coordinate1 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Midpoint reverse MQ displacement coordinate0 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Midpoint reverse PM squared sum with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Midpoint reverse PM coordinate1 square with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Midpoint reverse PM coordinate0 square with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Midpoint reverse PM displacement coordinate1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Midpoint reverse PM displacement coordinate0 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Midpoint ORIGINAL full squared length over four with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Midpoint ORIGINAL average coordinate1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Midpoint ORIGINAL endpoint sum coordinate1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Midpoint ORIGINAL average coordinate0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Midpoint ORIGINAL endpoint sum coordinate0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
• S006 · premise_and_result_audit — Recompute Length original PQ ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Length original PQ ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Length original PQ ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Length original PQ ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Length original PQ ORIGINAL Q minus P coordinate0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.

Result

(1/2, 23/2, 1061/2, 1061/2)

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.