HMMT 二月 2005 · 几何 · 第 4 题

HMMT February 2005 — Geometry — Problem 4

Let XY Z be a triangle with X = 60 and Y = 45 . A circle with center P passes through points A and B on side XY , C and D on side Y Z , and E and F on side ZX . 6 Suppose AB = CD = EF . Find XP Y in degrees.

解析

Let XY Z be a triangle with X = 60 and Y = 45 . A circle with center P passes through points A and B on side XY , C and D on side Y Z , and E and F on side ZX . 6 Suppose AB = CD = EF . Find XP Y in degrees. 1 Solution: 255 / 2 Since P AB , P CD , and P EF are all isosceles triangles with equal legs and equal bases, they are congruent. It follows that the heights of each are the same, so that P is equidistant from the sides of XY Z . Therefore, P is the incenter and therefore lies on 1 ◦ 1 6 6 6 6 the angle bisectors of XY Z . Thus Y XP = Y XZ = 30 and P Y X = ZY X = 2 2 ◦ ◦ ◦ 45 45 255 ◦ ◦ 6 . Therefore XP Y = 180 − 30 − = . 2 2 2