PUMaC 2023 · 几何（A 组） · 第 4 题

PUMaC 2023 — Geometry (Division A) — Problem 4

Let △ ABC be a triangle with AB = 4, BC = 6, and CA = 5. Let the angle bisector of ∠ BAC intersect BC at the point D and the circumcircle of △ ABC again at the point M ̸ = A . The perpendicular bisector of segment DM intersects the circle centered at M passing through B at two points, X and Y . Compute AX · AY .

解析

Let △ ABC be a triangle with AB = 4, BC = 6, and CA = 5. Let the angle bisector of ∠ BAC intersect BC at the point D and the circumcircle of △ ABC again at the point M ̸ = A . The perpendicular bisector of segment DM intersects the circle centered at M passing through B at two points, X and Y . Compute AX · AY . Proposed by Eric Shen Answer: 36 Note that AX = AY by symmetry and that AX = AM by inversion about M . In a 4-5-6 triangle we have the following relation between the angles: A = 2 C . Since AM subtends an A A 2 angle of + C and since + C = A , it follows that AM = BC = 6. Our answer is 6 = 36. 2 2