HMMT 二月 2024 · 几何 · 第 5 题

HMMT February 2024 — Geometry — Problem 5

Let ABCD be a convex trapezoid such that ∠ DAB = ∠ ABC = 90 , DA = 2 , AB = 3 , and BC = 8 . Let ω be a circle passing through A and tangent to segment CD at point T . Suppose that the center of ω lies on line BC . Compute CT .

解析

Let ABCD be a convex trapezoid such that ∠ DAB = ∠ ABC = 90 , DA = 2 , AB = 3 , and BC = 8 . Let ω be a circle passing through A and tangent to segment CD at point T . Suppose that the center of ω lies on line BC . Compute CT . Proposed by: Pitchayut Saengrungkongka √ √ Answer: 4 5 − 7 Solution: C T D ′ P A B A ′ Let A be the reflection of A across BC , and let P = AB ∩ CD . Then since the center of ω lies on BC , ′ 2 ′ we have that ω passes through A . Thus, by power of a point, P T = P A · P A . By similar triangles, we have P A P B P A P A + 3 = = ⇒ = = ⇒ P A = 1 , AD BC 2 8 √ √ √ ′ 2 2 and A P = 1 + 2 · 3 = 7 , so P T = 7 . But by the Pythagorean Theorem, P C = P B + BC = 4 5 , √ √ and since T lies on segment CD , it lies between C and P , so CT = 4 5 − 7 .