# Question 28892: Labeled Similarity at a Specified Positive Scale [Return to catalogue page 145](https://ocme.evemisslab.com/en/questions/page/145/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261006-advance-20261007-r06-similarity-foundations/#q-20261006-advance-20261007-r06-similarity-0017) 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. R0056-28892 · q-20261006-advance-20261007-r06-similarity-0017 ## Original question and givens Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(54, -4), B3=(20, -38), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=17/1. Does this scale fit every corresponding side? - G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(54, -4), B3=(20, -38), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=17/1. Does this scale fit every corresponding side? ## 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-similarity](https://ocme.evemisslab.com/en/mko/mko-similarity/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-similarity-specified-scale, specified_similarity] After derived = [ALL_fixed_corresponding_sides_at_given_forward_scale] Conditions: none Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions. Step identifiers and dependenciesRule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001 S002 · calculationBefore Similarity ORIGINAL scale numerator squared := mul(17, 17) After Similarity ORIGINAL scale numerator squared = 289 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Similarity ORIGINAL scale denominator squared := mul(1, 1) After Similarity ORIGINAL scale denominator squared = 1 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Similarity A determinant ORIGINAL point1 coordinate0 := sub(3, 0) After Similarity A determinant ORIGINAL point1 coordinate0 = 3 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Similarity A determinant ORIGINAL point1 coordinate1 := sub(0, 0) After Similarity A determinant ORIGINAL point1 coordinate1 = 0 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Similarity A determinant ORIGINAL point2 coordinate0 := sub(1, 0) After Similarity A determinant ORIGINAL point2 coordinate0 = 1 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Similarity A determinant ORIGINAL point2 coordinate1 := sub(2, 0) After Similarity A determinant ORIGINAL point2 coordinate1 = 2 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Similarity A determinant ORIGINAL first cross product := mul(3, 2) After Similarity A determinant ORIGINAL first cross product = 6 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Similarity A determinant ORIGINAL second cross product := mul(0, 1) After Similarity A determinant ORIGINAL second cross product = 0 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Similarity A determinant ORIGINAL cross difference := sub(6, 0) After Similarity A determinant ORIGINAL cross difference = 6 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore source roles = [Similarity A, [[0, 0], [3, 0], [1, 2]]] After derived = [6, Yes] Conditions: none Require an ACTUAL nondegenerate triangle via the nonzero cross product of two basepoint vectors.Retain original A1,A2,A3 or B1,B2,B3 labels;do not search another correspondence. Step identifiers and dependenciesRule: Similarity_ACTUAL_proper_ordered_triangle_no_relabel · Uses: G001, S010 S012 · calculationBefore Similarity A side12 ORIGINAL coordinate0 difference := sub(3, 0) After Similarity A side12 ORIGINAL coordinate0 difference = 3 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Similarity A side12 ORIGINAL coordinate1 difference := sub(0, 0) After Similarity A side12 ORIGINAL coordinate1 difference = 0 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Similarity A side12 ORIGINAL coordinate0 square := mul(3, 3) After Similarity A side12 ORIGINAL coordinate0 square = 9 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Similarity A side12 ORIGINAL coordinate1 square := mul(0, 0) After Similarity A side12 ORIGINAL coordinate1 square = 0 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Similarity A side12 ORIGINAL squared length := sum(9, 0) After Similarity A side12 ORIGINAL squared length = 9 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Similarity A side13 ORIGINAL coordinate0 difference := sub(1, 0) After Similarity A side13 ORIGINAL coordinate0 difference = 1 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Similarity A side13 ORIGINAL coordinate1 difference := sub(2, 0) After Similarity A side13 ORIGINAL coordinate1 difference = 2 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Similarity A side13 ORIGINAL coordinate0 square := mul(1, 1) After Similarity A side13 ORIGINAL coordinate0 square = 1 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Similarity A side13 ORIGINAL coordinate1 square := mul(2, 2) After Similarity A side13 ORIGINAL coordinate1 square = 4 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Similarity A side13 ORIGINAL squared length := sum(1, 4) After Similarity A side13 ORIGINAL squared length = 5 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Similarity A side23 ORIGINAL coordinate0 difference := sub(1, 3) After Similarity A side23 ORIGINAL coordinate0 difference = -2 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Similarity A side23 ORIGINAL coordinate1 difference := sub(2, 0) After Similarity A side23 ORIGINAL coordinate1 difference = 2 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Similarity A side23 ORIGINAL coordinate0 square := mul(-2, -2) After Similarity A side23 ORIGINAL coordinate0 square = 4 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Similarity A side23 ORIGINAL coordinate1 square := mul(2, 2) After Similarity A side23 ORIGINAL coordinate1 square = 4 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Similarity A side23 ORIGINAL squared length := sum(4, 4) After Similarity A side23 ORIGINAL squared length = 8 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Similarity B determinant ORIGINAL point1 coordinate0 := sub(54, 3) After Similarity B determinant ORIGINAL point1 coordinate0 = 51 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Similarity B determinant ORIGINAL point1 coordinate1 := sub(-4, -4) After Similarity B determinant ORIGINAL point1 coordinate1 = 0 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore Similarity B determinant ORIGINAL point2 coordinate0 := sub(20, 3) After Similarity B determinant ORIGINAL point2 coordinate0 = 17 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore Similarity B determinant ORIGINAL point2 coordinate1 := sub(-38, -4) After Similarity B determinant ORIGINAL point2 coordinate1 = -34 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore Similarity B determinant ORIGINAL first cross product := mul(51, -34) After Similarity B determinant ORIGINAL first cross product = -1734 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore Similarity B determinant ORIGINAL second cross product := mul(0, 17) After Similarity B determinant ORIGINAL second cross product = 0 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Similarity B determinant ORIGINAL cross difference := sub(-1734, 0) After Similarity B determinant ORIGINAL cross difference = -1734 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · impliesBefore source roles = [Similarity B, [[3, -4], [54, -4], [20, -38]]] After derived = [-1734, Yes] Conditions: none Require an ACTUAL nondegenerate triangle via the nonzero cross product of two basepoint vectors.Retain original A1,A2,A3 or B1,B2,B3 labels;do not search another correspondence. Step identifiers and dependenciesRule: Similarity_ACTUAL_proper_ordered_triangle_no_relabel · Uses: G001, S033 S035 · calculationBefore Similarity B side12 ORIGINAL coordinate0 difference := sub(54, 3) After Similarity B side12 ORIGINAL coordinate0 difference = 51 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore Similarity B side12 ORIGINAL coordinate1 difference := sub(-4, -4) After Similarity B side12 ORIGINAL coordinate1 difference = 0 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore Similarity B side12 ORIGINAL coordinate0 square := mul(51, 51) After Similarity B side12 ORIGINAL coordinate0 square = 2601 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore Similarity B side12 ORIGINAL coordinate1 square := mul(0, 0) After Similarity B side12 ORIGINAL coordinate1 square = 0 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore Similarity B side12 ORIGINAL squared length := sum(2601, 0) After Similarity B side12 ORIGINAL squared length = 2601 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore Similarity B side13 ORIGINAL coordinate0 difference := sub(20, 3) After Similarity B side13 ORIGINAL coordinate0 difference = 17 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore Similarity B side13 ORIGINAL coordinate1 difference := sub(-38, -4) After Similarity B side13 ORIGINAL coordinate1 difference = -34 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore Similarity B side13 ORIGINAL coordinate0 square := mul(17, 17) After Similarity B side13 ORIGINAL coordinate0 square = 289 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore Similarity B side13 ORIGINAL coordinate1 square := mul(-34, -34) After Similarity B side13 ORIGINAL coordinate1 square = 1156 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore Similarity B side13 ORIGINAL squared length := sum(289, 1156) After Similarity B side13 ORIGINAL squared length = 1445 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore Similarity B side23 ORIGINAL coordinate0 difference := sub(20, 54) After Similarity B side23 ORIGINAL coordinate0 difference = -34 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore Similarity B side23 ORIGINAL coordinate1 difference := sub(-38, -4) After Similarity B side23 ORIGINAL coordinate1 difference = -34 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore Similarity B side23 ORIGINAL coordinate0 square := mul(-34, -34) After Similarity B side23 ORIGINAL coordinate0 square = 1156 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore Similarity B side23 ORIGINAL coordinate1 square := mul(-34, -34) After Similarity B side23 ORIGINAL coordinate1 square = 1156 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore Similarity B side23 ORIGINAL squared length := sum(1156, 1156) After Similarity B side23 ORIGINAL squared length = 2312 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore Similarity fixed side0 ORIGINAL Bsquare times q squared := mul(2601, 1) After Similarity fixed side0 ORIGINAL Bsquare times q squared = 2601 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore Similarity fixed side0 ORIGINAL Asquare times p squared := mul(9, 289) After Similarity fixed side0 ORIGINAL Asquare times p squared = 2601 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · impliesBefore source roles = [0, 9, 2601, 17, 1] After derived = [2601, 2601, Yes] Conditions: none Use the given FORWARD scale from A to B.Positive lengths give B²*q²=A²*p² IFF B/A=p/q.Check EVERY side12,13,23,not merely the first. Step identifiers and dependenciesRule: Similarity_EACH_fixed_corresponding_squared_side_equality · Uses: G001, S051 S053 · calculationBefore Similarity fixed side1 ORIGINAL Bsquare times q squared := mul(1445, 1) After Similarity fixed side1 ORIGINAL Bsquare times q squared = 1445 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · calculationBefore Similarity fixed side1 ORIGINAL Asquare times p squared := mul(5, 289) After Similarity fixed side1 ORIGINAL Asquare times p squared = 1445 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · impliesBefore source roles = [1, 5, 1445, 17, 1] After derived = [1445, 1445, Yes] Conditions: none Use the given FORWARD scale from A to B.Positive lengths give B²*q²=A²*p² IFF B/A=p/q.Check EVERY side12,13,23,not merely the first. Step identifiers and dependenciesRule: Similarity_EACH_fixed_corresponding_squared_side_equality · Uses: G001, S054 S056 · calculationBefore Similarity fixed side2 ORIGINAL Bsquare times q squared := mul(2312, 1) After Similarity fixed side2 ORIGINAL Bsquare times q squared = 2312 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · calculationBefore Similarity fixed side2 ORIGINAL Asquare times p squared := mul(8, 289) After Similarity fixed side2 ORIGINAL Asquare times p squared = 2312 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · impliesBefore source roles = [2, 8, 2312, 17, 1] After derived = [2312, 2312, Yes] Conditions: none Use the given FORWARD scale from A to B.Positive lengths give B²*q²=A²*p² IFF B/A=p/q.Check EVERY side12,13,23,not merely the first. Step identifiers and dependenciesRule: Similarity_EACH_fixed_corresponding_squared_side_equality · Uses: G001, S057 S059 · impliesBefore source roles = [[[0, 0], [3, 0], [1, 2]], [[3, -4], [54, -4], [20, -38]], 17, 1] After derived = [[Yes, Yes, Yes], Yes, Yes] Conditions: none For proper triangles,ALL three corresponding sides at the same positive scale give SSS similarity,including a reflected copy.No rejects THIS correspondence and forward scale,not every possible ratio or permutation. Step identifiers and dependenciesRule: Similarity_SSS_ALL_three_FIXED_correspondence_and_given_scale · Uses: G001, S058 ## Reverse checks - S059 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. - S058 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. - S057 · premise_and_result_audit — Recompute Similarity fixed side2 ORIGINAL Asquare times p squared with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute Similarity fixed side2 ORIGINAL Bsquare times q squared with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. - S054 · premise_and_result_audit — Recompute Similarity fixed side1 ORIGINAL Asquare times p squared with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute Similarity fixed side1 ORIGINAL Bsquare times q squared with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. - S051 · premise_and_result_audit — Recompute Similarity fixed side0 ORIGINAL Asquare times p squared with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Similarity fixed side0 ORIGINAL Bsquare times q squared with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Similarity B side23 ORIGINAL squared length with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Similarity B side23 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute Similarity B side23 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute Similarity B side23 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Similarity B side23 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute Similarity B side13 ORIGINAL squared length with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Similarity B side13 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute Similarity B side13 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute Similarity B side13 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Similarity B side13 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Similarity B side12 ORIGINAL squared length with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Similarity B side12 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Similarity B side12 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Similarity B side12 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Similarity B side12 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. - S033 · premise_and_result_audit — Recompute Similarity B determinant ORIGINAL cross difference with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Similarity B determinant ORIGINAL second cross product with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Similarity B determinant ORIGINAL first cross product with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Similarity B determinant ORIGINAL point2 coordinate1 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Similarity B determinant ORIGINAL point2 coordinate0 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Similarity B determinant ORIGINAL point1 coordinate1 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Similarity B determinant ORIGINAL point1 coordinate0 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Similarity A side23 ORIGINAL squared length with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Similarity A side23 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Similarity A side23 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Similarity A side23 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Similarity A side23 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Similarity A side13 ORIGINAL squared length with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Similarity A side13 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Similarity A side13 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Similarity A side13 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Similarity A side13 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Similarity A side12 ORIGINAL squared length with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Similarity A side12 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Similarity A side12 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Similarity A side12 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Similarity A side12 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. - S010 · premise_and_result_audit — Recompute Similarity A determinant ORIGINAL cross difference with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Similarity A determinant ORIGINAL second cross product with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Similarity A determinant ORIGINAL first cross product with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Similarity A determinant ORIGINAL point2 coordinate1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Similarity A determinant ORIGINAL point2 coordinate0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Similarity A determinant ORIGINAL point1 coordinate1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Similarity A determinant ORIGINAL point1 coordinate0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Similarity ORIGINAL scale denominator squared with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Similarity ORIGINAL scale numerator squared with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. ## Result Yes 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: e3cea099d694b859110419193803308b142992c27f664f19af85e22629cff106 - candidate record SHA-256: 5a2f878a8ded110342275c1d2922ba5be134b0862180203402cea9044c9c7f9c - semantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8 - question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261006-advance-20261007-r06-similarity-0017.json)