HMMT 十一月 2014 · GEN 赛 · 第 6 题

HMMT November 2014 — GEN Round — Problem 6

Let ABC be a triangle with AB = 5, AC = 4, BC = 6. The angle bisector of C intersects side AB at X . Points M and N are drawn on sides BC and AC , respectively, such that XM ‖ AC and XN ‖ BC . Compute the length M N .

解析

Let ABC be a triangle with AB = 5, AC = 4, BC = 6. The angle bisector of C intersects side AB at X . Points M and N are drawn on sides BC and AC , respectively, such that XM ‖ AC and XN ‖ BC . Compute the length M N . √ 3 14 Answer: By Stewart’s Theorem on the angle bisector, 5 ( ) 2 AB 2 CX = AC · BC 1 − AC + BC Thus, ( ) 2 5 2 CX = 4 · 6 1 − = 18 10 Since XM ‖ AC and XN ‖ BC , we produce equal angles. So, by similar triangles, XM = XN = 4 · 6 12 = . 10 5 Moreover, triangles M CX and N CX are congruent isosceles triangles with vertices M and N , respec- tively. Since CX is an angle bisector, then CX and M N are perpendicular bisectors of each other. Therefore, ( ) 2 12 126 2 2 2 M N = 4( XN − ( CX/ 2) ) = 4 · − 18 = 5 25 and √ 3 14 M N = 5