HMMT 十一月 2011 · THM 赛 · 第 9 题

英文解析

9. [ 6 ] Let ABC be a triangle with AB = 13, BC = 14, and CA = 15. Let D be the foot of the altitudefrom A to BC . The inscribed circles of triangles ABD and ACD are tangent to AD at P and Q ,
   respectively, and are tangent to BC at X and Y , respectively. Let P X and QY meet at Z . Determinethe area of triangle XY Z .
   Answer: First, note that AD = 12 , BD = 5 , CD = 9.25
   By equal tangents, we get that P D = DX , so P DX is isosceles. Because D is a right angle, we get that 4
   ° °
   ∠ P XD = 45 . Similarly, ∠ XY Z = 45 , so XY Z is an isosceles right triangle with hypotenuse XY .
   1 1
   However, by tangents to the incircle, we get that XD = (12+5 − 13) = 2 and Y D = (12+9 − 15) = 3.
   2 2
   1 1 25
   2 2
   Hence, the area of the XYZ is ( XY ) = (2 + 3) = .
   4 4 4
   PZQA
   B X D Y C