HMMT 十一月 2023 · 冲刺赛 · 第 17 题

HMMT November 2023 — Guts Round — Problem 17

[10] Let ABC be an equilateral triangle of side length 15. Let A and B be points on side AB , A b a c and C be points on side AC , and B and C be points on side BC such that △ AA A , △ BB B , and a c b b c c a △ CC C are equilateral triangles with side lengths 3, 4, and 5, respectively. Compute the radius of the a b circle tangent to segments A A , B B , and C C . b c a c a b

解析

[10] Let ABC be an equilateral triangle of side length 15. Let A and B be points on side AB , A b a c and C be points on side AC , and B and C be points on side BC such that △ AA A , △ BB B , a c b b c c a and △ CC C are equilateral triangles with side lengths 3, 4, and 5, respectively. Compute the radius a b of the circle tangent to segments A A , B B , and C C . b c a c a b Proposed by: Pitchayut Saengrungkongka √ Answer: 3 3. Solution: A A A b c Z Y C a B a B C B C c b X Let △ XY Z be the triangle formed by lines A A , B B , and C C . Then, the desired circle is the b c a c a b incircle of △ XY Z , which is equilateral. We have Y Z = Y A + A A + A Z c c b b = A C + A A + A B c a c b b a = (15 − 3 − 5) + 3 + (15 − 3 − 4) = 18 , √ 1 √ and so the inradius is · 18 = 3 3. 2 3