HMMT 十一月 2011 · THM 赛 · 第 9 题

HMMT November 2011 — THM Round — Problem 9

[ 6 ] Let ABC be a triangle with AB = 13, BC = 14, and CA = 15. Let D be the foot of the altitude from 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 . Determine the area of triangle XY Z .

解析

[ 6 ] Let ABC be a triangle with AB = 13, BC = 14, and CA = 15. Let D be the foot of the altitude from 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 . Determine the area of triangle XY Z . 25 Answer: First, note that AD = 12 , BD = 5 , CD = 9. 4 By equal tangents, we get that P D = DX , so P DX is isosceles. Because D is a right angle, we get that ◦ ◦ ∠ 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 A Q Z P B X D Y C