HMMT 二月 2014 · 冲刺赛 · 第 30 题

英文解析

30. [ 20 ] Let ABC be a triangle with circumcenter O , incenter I , ∠ B = 45 , and OI ‖ BC . Find cos ∠ C .
    √
    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 =2
    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