HMMT 十一月 2011 · GEN 赛 · 第 2 题

英文解析

2. [ 3 ] Let ABC be a triangle, and let D , E , and F be the midpoints of sides BC , CA , and AB , respectively.
   ° °
   Let the angle bisectors of ∠ F DE and ∠ F BD meet at P . Given that ∠ BAC = 37 and ∠ CBA = 85 ,
   determine the degree measure of ∠ BP D .
   °
   Answer: 61 Because D, E, F are midpoints, we have ABC ∼ DEF . Furthermore, we know that
   F D ‖ AC and DE ‖ AB , so we have
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   ∠ BDF = ∠ BCA = 180 − 37 − 85 = 58 .
   °
   Also, ∠ F DE = ∠ BAC = 37 . Hence, we have
   ( )
   ° °
   85 37
   ° ° ° °
   ∠ BP D = 180 − ∠ P BD − ∠ P DB = 180 − − + 58 = 61 .
   2 2
   37A
   58PFE
   B D C85