HMMT 二月 2017 · 几何 · 第 9 题

英文解析

9. Let ABC be a triangle, and let BCDE, CAF G, ABHI be squares that do not overlap the trianglewith centers X, Y, Z respectively. Given that AX = 6, BY = 7, and CZ = 8, find the area of triangle
   XY Z .
   Proposed by: Sam Korsky
   √
   21 15
   Answer:
   By the degenerate case of Von Aubel’s Theorem we have that Y Z = AX = 6 and ZX = BY = 7 and 4
   √
   21 15
   XY = CZ = 8 so it suffices to find the area of a 6 − 7 − 8 triangle which is given by .
   To prove that AX = Y Z , note that by Lo c we get 4
   2 2
   b c
   Y X = + + bc sin ∠ A2
   2 2
   2 anda
   2 2
   AX = b + − ab (cos ∠ C − sin ∠ C )
   a 22 = c + − ac (cos ∠ B − sin ∠ B )2
   2 22
   b + c + a ( b sin ∠ C + c sin ∠ B )
   2=
   2 2
   b c = + + ah
   2 2
   where h is the length of the A -altitude of triangle ABC . In these calculations we used the well-knownfact that b cos ∠ C + c cos ∠ B = a which can be easily seen by drawing in the A -altitude. Then sincebc sin ∠ A and ah both equal twice the area of triangle ABC , we are done.