HMMT 十一月 2015 · 冲刺赛 · 第 25 题

HMMT November 2015 — Guts Round — Problem 25

[ 13 ] Let ABC be a triangle that satisfies AB = 13 , BC = 14 , AC = 15 . Given a point P in the plane, let P , P , P be the reflections of A, B, C across P . Call P good if the circumcircle of P P P A B C A B C intersects the circumcircle of ABC at exactly 1 point. The locus of good points P encloses a region S . Find the area of S .

解析

[ 13 ] Let ABC be a triangle that satisfies AB = 13 , BC = 14 , AC = 15 . Given a point P in the plane, let P , P , P be the reflections of A, B, C across P . Call P good if the circumcircle of P P P A B C A B C intersects the circumcircle of ABC at exactly 1 point. The locus of good points P encloses a region S . Find the area of S . Proposed by: Yang Liu 4225 Answer: π 64 By the properties of reflection, the circumradius of P P P equals the circumradius of ABC . There- A B C fore, the circumcircle of P P P must be externally tangent to the circumcircle of ABC . Now it’s A B C easy to see that the midpoint of the 2 centers of ABC and P P P lies on the circumcircle of ABC. A B C So the locus of P is simply the circumcircle of ABC . abc 13 · 14 · 15 65 Since [ ABC ] = , we find the circumradius is R = = , so the enclosed region has area 4 R 84 · 4 8 4225 π . 64