HMMT 二月 2014 · 冲刺赛 · 第 30 题

HMMT February 2014 — Guts Round — Problem 30

[ 20 ] Let ABC be a triangle with circumcenter O , incenter I , \ B = 45 , and OI k BC . Find cos \ C .

解析

[ 20 ] Let ABC be a triangle with circumcenter O , incenter I , ∠ B = 45 , and OI ‖ BC . Find cos ∠ C . √ 2 Answer: 1 − Let M be the midpoint of BC , and D the foot of the perpendicular of I with BC . 2 Because OI || BC , we have OM = ID . Since ∠ BOC = 2 ∠ A , the length of OM is OA cos ∠ BOM = OA cos A = R cos A , and the length of ID is r , where R and r are the circumradius and inradius of 4 ABC , respectively. Thus, r = R cos A , so 1 + cos A = ( R + r ) /R . By Carnot’s theorem, ( R + r ) /R = cos A + cos B + cos C , √ √ 2 2 so we have cos B + cos C = 1. Since cos B = , we have cos C = 1 − . 2 2