HMMT 二月 2013 · 几何 · 第 5 题
HMMT February 2013 — Geometry — Problem 5
题目详情
英文原题
In triangle ABC ,∠A= 45◦andMis the midpoint of BC.AM intersects the circumcircle of ABC for the second time at D, and AM = 2MD . Find cos ∠AOD , where Ois the circumcenter of ABC .
解析
英文解析
- In triangle ABC , ∠ A = 45 and M is the midpoint of BC . AM intersects the circumcircle of ABCfor 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
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