HMMT 十一月 2014 · GEN 赛 · 第 6 题

英文解析

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 .
   √
   3 14
   Answer: By Stewart’s Theorem on the angle bisector,
   ( )5
   AB2
   CX = AC · BC 1 −2
   AC + BC
   Thus,
   ( )
   CX = 4 · 6 1 − = 18252
   Since XM ‖ AC and XN ‖ BC , we produce equal angles. So, by similar triangles, XM = XN =10
   4 · 6 12 = .
   10 5
   Moreover, triangles M CX and N CX are congruent isosceles triangles with vertices M and N , respectively. Since CX is an angle bisector, then CX and M N are perpendicular bisectors of each other.
   Therefore,
   ( )
   12 1262
   2 2 2
   M N = 4( XN − ( CX/ 2) ) = 4 · − 18 =
   5 25
   √and
   3 14
   M N = 5