HMMT 二月 2016 · 冲刺赛 · 第 10 题

HMMT February 2016 — Guts Round — Problem 10

[ 7 ] Let ABC be a triangle with AB = 13, BC = 14, CA = 15. Let O be the circumcenter of ABC . Find the distance between the circumcenters of triangles AOB and AOC . !

解析

[ 7 ] Let ABC be a triangle with AB = 13, BC = 14, CA = 15. Let O be the circumcenter of ABC . Find the distance between the circumcenters of triangles AOB and AOC . Proposed by: Evan Chen 91 Answer: 6 Let S, T be the intersections of the tangents to the circumcircle of ABC at A, C and at A, B respectively. Note that ASCO is cyclic with diameter SO , so the circumcenter of AOC is the midpoint of OS , and 1 similarly for the other side. So the length we want is ST . The circumradius R of ABC can be 2 abc 65 computed by Heron’s formula and K = , giving R = . A few applications of the Pythagorean 4 R 8 91 65 39 theorem and similar triangles gives AT = , AS = , so the answer is . 6 2 6 !