HMMT 二月 2013 · 几何 · 第 5 题

HMMT February 2013 — Geometry — Problem 5

In triangle ABC , ∠ A = 45 and M is the midpoint of BC . AM intersects the circumcircle of ABC for the second time at D , and AM = 2 M D . Find cos ∠ AOD , where O is the circumcenter of ABC . ◦

解析

In triangle ABC , ∠ A = 45 and M is the midpoint of BC . AM intersects the circumcircle of ABC for the second time at D , and AM = 2 M D . Find cos ∠ AOD , where O is the circumcenter of ABC . √ 1 ◦ ◦ Answer: − ∠ BAC = 45 , so ∠ BOC = 90 . If the radius of the circumcircle is r , BC = 2 r , 8 √ 2 1 and BM = CM = r . By power of a point, BM · CM = AM · DM , so AM = r and DM = r , and 2 2 3 1 AD = r . Using the law of cosines on triangle AOD gives cos ∠ AOD = − . 2 8 Geometry Test ◦