# Question 2972: Membership in a Circle Boundary [Return to catalogue page 15](https://ocme.evemisslab.com/en/questions/page/15/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261005-advance-20261006-r05-circle-foundations/#q-20261005-advance-20261006-r05-circle-0022) 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. R0004-02972 · q-20261005-advance-20261006-r05-circle-0022 ## Original question and givens Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(9, -16) and radius r=20≥0. For P=(29, -16), is P a member of the circle boundary, not merely of the filled disk? - G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(9, -16) and radius r=20≥0. For P=(29, -16), is P a member of the circle boundary, not merely of the filled disk? ## 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-circle](https://ocme.evemisslab.com/en/mko/mko-circle/) ## Step-by-step derivation S001 · impliesBefore orthonormal equal-length units;r=20≥0 After C={P:distance(P,O)=r} is a boundary,not a filled disk Conditions: none The circle is the boundary at distance exactly r from the center,not the filled disk.For r=0 use the degenerate singleton{O},with no inside points and diameter length0. Step identifiers and dependenciesRule: circle_boundary_convention · Uses: G001 S002 · calculationBefore radius squared := square(20) After radius squared = 400 Conditions: none The square of the nonnegative radius is the distance-comparison threshold. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore coordinate difference0 := sub(29, 9) After coordinate difference0 = 20 Conditions: none Subtract the center coordinate from the point coordinate;do not square absolute coordinates instead. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore coordinate square0 := square(20) After coordinate square0 = 400 Conditions: none Square this coordinate difference;a negative difference still contributes a nonnegative square. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore coordinate difference1 := sub(-16, -16) After coordinate difference1 = 0 Conditions: none Subtract the center coordinate from the point coordinate;do not square absolute coordinates instead. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore coordinate square1 := square(0) After coordinate square1 = 0 Conditions: none Square this coordinate difference;a negative difference still contributes a nonnegative square. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore squared distance := sum(400, 0) After squared distance = 400 Conditions: none In an orthonormal equal-unit frame,the Pythagorean relation adds the squared differences. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore d²=400;r²=400 After requested classification=Yes Conditions: none Distance and radius are nonnegative,so squaring preserves their order.Equality means boundary,strictly smaller means inside,and larger means outside.No rounded square root is needed. Step identifiers and dependenciesRule: nonnegative_squared_distance_comparison · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute squared distance with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute coordinate square1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute coordinate difference1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute coordinate square0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute coordinate difference0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute radius squared with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## 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: ffc7e631f96c30f45ec4a5e02190aa6a0f0fc8749cbcb8d0442ad8e48e515372 - candidate record SHA-256: 2673c5a50f1fd707b4d1facd51a89123a0eb28606d9d45d870fd3c4539862c07 - semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56 - question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261005-advance-20261006-r05-circle-0022.json)