HMMT 十一月 2011 · GEN 赛 · 第 2 题

HMMT November 2011 — GEN Round — Problem 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 .

解析

[ 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 ◦ ∠ 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 A 37 E F P 58 85 B D C