HMMT 二月 2020 · 冲刺赛 · 第 17 题

HMMT February 2020 — Guts Round — Problem 17

[10] Let ABC be a triangle with incircle tangent to the perpendicular bisector of BC . If BC = AE = 20, where E is the point where the A -excircle touches BC , then compute the area of 4 ABC .

解析

[10] Let ABC be a triangle with incircle tangent to the perpendicular bisector of BC . If BC = AE = 20, where E is the point where the A -excircle touches BC , then compute the area of 4 ABC . Proposed by: Tristan Shin √ Answer: 100 2 Solution: Let the incircle and BC touch at D , the incircle and perpendicular bisector touch at X , Y be the point opposite D on the incircle, and M be the midpoint of BC . Recall that A , Y , and E are ◦ collinear by homothety at A . Additionally, we have M D = M X = M E so ∠ DXY = ∠ DXE = 90 . ◦ Therefore E , X , and Y are collinear. Since M X ⊥ BC , we have ∠ AEB = 45 . The area of ABC is √ 1 BC · AE · sin ∠ AEB = 100 2 . 2 A Y X I B D M E C