HMMT 二月 2026 · 几何 · 第 4 题

HMMT February 2026 — Geometry — Problem 4

Let ABC be a triangle with ∠ BAC = 90 . Points X and Y are such that B , X , Y , and C lie on segment BC in that order, BX = 4 , XY = 5 , and Y C = 3 . Let T be a point lying on segment AC such that T A = T X = T Y = ℓ for some ℓ . Compute ℓ .

解析

Let ABC be a triangle with ∠ BAC = 90 . Points X and Y are such that B , X , Y , and C lie on segment BC in that order, BX = 4 , XY = 5 , and Y C = 3 . Let T be a point lying on segment AC such that T A = T X = T Y = ℓ for some ℓ . Compute ℓ . Proposed by: Jason Mao √ 7 Answer: 3 3 Solution: ©2026 HMMT A T Z B C X Y Draw circle ω with center T and radius ℓ , which passes through points A, X, Y . Since ω is tangent to BA at A , taking the power of B with respect to ω yields 2 BA = BX · BY = 4 · 9 = ⇒ BA = 6 . √ √ 2 2 Thus we get AC = BC − BA = 6 3 . Let Z be the reflection of A across T , which is on ω . Taking the power of C with respect to ω yields √ √ CY · CX = CZ · CA = ⇒ 3 · 8 = (6 3 − 2 ℓ ) · 6 3 √ 7 3 Solving this equation, we get that ℓ = . 3