HMMT 十一月 2015 · GEN 赛 · 第 7 题

HMMT November 2015 — GEN Round — Problem 7

Let 4 ABC be a right triangle with right angle C . Let I be the incenter of ABC , and let M lie on AC and N on BC , respectively, such that M, I, N are collinear and M N is parallel to AB . If AB = 36 and the perimeter of CM N is 48, find the area of ABC .

解析

Let 4 ABC be a right triangle with right angle C . Let I be the incenter of ABC , and let M lie on AC and N on BC , respectively, such that M, I, N are collinear and M N is parallel to AB . If AB = 36 and the perimeter of CM N is 48, find the area of ABC . Proposed by: Alexander Katz Answer: 252 Note that ∠ M IA = ∠ BAI = ∠ CAI , so M I = M A . Similarly, N I = N B . As a result, CM + M N + N C = CM + M I + N I + N C = CM + M A + N B + N C = AC + BC = 48. Furthermore, 2 2 2 2 2 2 2 2 AC + BC = 36 . As a result, we have AC +2 AC · BC + BC = 48 , so 2 AC · BC = 48 − 36 = 12 · 84, AC · BC and so = 3 · 84 = 252 . 2