# Question 30194: Exact Midpoint and Two Squared Segment Lengths [Return to catalogue page 151](https://ocme.evemisslab.com/en/questions/page/151/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261012-r06-euclidean-length-foundations/#q-20261008-advance-20261012-r06-euclidean-length-0069) 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. R0059-30194 · q-20261008-advance-20261012-r06-euclidean-length-0069 ## Original question and givens Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-18, 1], Q=[19, 20], 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=[-18, 1], Q=[19, 20], 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](https://ocme.evemisslab.com/en/mko/mko-euclidean-length/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density12-length-midpoint-equidistance, density12_length_midpoint_equidistance, integer_euclidean_plane] After derived = [[midpoint_x, midpoint_y, SQUARED_PM, SQUARED_MQ]] Conditions: none Retain the ORIGINAL bank full operation prefix,domain and ordered answer roles.Use orthogonal_equal_unit only where a frame is declared,and degrees where an angle unit is declared.Integer fractions retain/1.Do not invent parameters or alter original bilingual sources. Step identifiers and dependenciesRule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001 S002 · calculationBefore Length original PQ ORIGINAL Q minus P coordinate0 := sub(19, -18) After Length original PQ ORIGINAL Q minus P coordinate0 = 37 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Length original PQ ORIGINAL Q minus P coordinate1 := sub(20, 1) After Length original PQ ORIGINAL Q minus P coordinate1 = 19 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Length original PQ ORIGINAL square coordinate0 := mul(37, 37) After Length original PQ ORIGINAL square coordinate0 = 1369 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Length original PQ ORIGINAL square coordinate1 := mul(19, 19) After Length original PQ ORIGINAL square coordinate1 = 361 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Length original PQ ORIGINAL squared norm sum := sum(1369, 361) After Length original PQ ORIGINAL squared norm sum = 1730 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [Length original PQ, [-18, 1], [19, 20]] After derived = [[37, 19], 1730] Conditions: none In the SAME equal-unit orthogonal L2 frame,the directed displacement is Q-P with signs retained.The sum of coordinate squares is the nonnegative SQUARED length.Derived coordinates may exceed SAFE without being revalidated as raw inputs. Step identifiers and dependenciesRule: Euclidean_DIRECTED_displacement_then_SQUARED_nonnegative_norm · Uses: G001, S006 S008 · calculationBefore Midpoint ORIGINAL endpoint sum coordinate0 := sum(-18, 19) After Midpoint ORIGINAL endpoint sum coordinate0 = 1 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Midpoint ORIGINAL average coordinate0 := div(1, 2) After Midpoint ORIGINAL average coordinate0 = 1/2 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Midpoint ORIGINAL endpoint sum coordinate1 := sum(1, 20) After Midpoint ORIGINAL endpoint sum coordinate1 = 21 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Midpoint ORIGINAL average coordinate1 := div(21, 2) After Midpoint ORIGINAL average coordinate1 = 21/2 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Midpoint ORIGINAL full squared length over four := div(1730, 4) After Midpoint ORIGINAL full squared length over four = 865/2 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Midpoint reverse PM displacement coordinate0 := sub(1/2, -18) After Midpoint reverse PM displacement coordinate0 = 37/2 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Midpoint reverse PM displacement coordinate1 := sub(21/2, 1) After Midpoint reverse PM displacement coordinate1 = 19/2 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Midpoint reverse PM coordinate0 square := mul(37/2, 37/2) After Midpoint reverse PM coordinate0 square = 1369/4 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Midpoint reverse PM coordinate1 square := mul(19/2, 19/2) After Midpoint reverse PM coordinate1 square = 361/4 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Midpoint reverse PM squared sum := sum(1369/4, 361/4) After Midpoint reverse PM squared sum = 865/2 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Midpoint reverse MQ displacement coordinate0 := sub(19, 1/2) After Midpoint reverse MQ displacement coordinate0 = 37/2 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Midpoint reverse MQ displacement coordinate1 := sub(20, 21/2) After Midpoint reverse MQ displacement coordinate1 = 19/2 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Midpoint reverse MQ coordinate0 square := mul(37/2, 37/2) After Midpoint reverse MQ coordinate0 square = 1369/4 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Midpoint reverse MQ coordinate1 square := mul(19/2, 19/2) After Midpoint reverse MQ coordinate1 square = 361/4 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Midpoint reverse MQ squared sum := sum(1369/4, 361/4) After Midpoint reverse MQ squared sum = 865/2 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore source roles = [[-18, 1], [19, 20], [1/2, 21/2], 1730] After derived = [865/2, 865/2, 865/2] Conditions: none Average EACH coordinate to obtain M,possibly half-integral.Both M-P and Q-M are half the original displacement,so EACH squared length is N/4,also recomputed separately.These are SQUARES,not square roots;P=Q is legal. Step identifiers and dependenciesRule: Length_midpoint_TWO_HALFsegment_SQUARES_not_distances · Uses: G001, S022 ## 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, 21/2, 865/2, 865/2) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: e31e2c41a926ab348ada3e6fb5ed0ba2320f55ec5119b788034df94a84fa7484 - candidate record SHA-256: 0a8d6923ff37c238a40c954b6222e79f58477ecbfaef522e155bb393e2e7e7d6 - semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5 - question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r06-euclidean-length-0069.json)