HMMT 二月 2023 · 几何 · 第 5 题

HMMT February 2023 — Geometry — Problem 5

Let ABC be a triangle with AB = 13, BC = 14, and CA = 15. Suppose P QRS is a square such that P and R lie on line BC , Q lies on line CA , and S lies on line AB . Compute the side length of this square. ◦ ◦

解析

Let ABC be a triangle with AB = 13, BC = 14, and CA = 15. Suppose P QRS is a square such that P and R lie on line BC , Q lies on line CA , and S lies on line AB . Compute the side length of this square. Proposed by: Pitchayut Saengrungkongka √ Answer: 42 2 Solution: Q A B C M X R P ′ A S ′ ′ Let A be the reflection of A across BC . Since Q and S are symmetric across BC , we get that Q ∈ BA , ′ ′ S ∈ CA . Now, let X and M be the midpoints of AA and P R . Standard altitude computation gives ′ BX = 5, CX = 9, AX = 12. Moreover, from similar triangles, CX : CY = AA : P R = BX : BM , so 12 BM : CM = 5 : 9, so we easily get that BM = 35 / 2. Now, P M = · BY = 42, so the side length is 9 √ 42 2. ◦ ◦