HMMT 二月 2006 · 几何 · 第 6 题
HMMT February 2006 — Geometry — Problem 6
题目详情
- A circle of radius t is tangent to the hypotenuse, the incircle, and one leg of an isosceles π right triangle with inradius r = 1 + sin . Find rt . 8
解析
- A circle of radius t is tangent to the hypotenuse, the incircle, and one leg of an isosceles π right triangle with inradius r = 1 + sin . Find rt . 8 √ 2 + 2 Answer: 4 Solution: The distance between the point of tangency of the two circles and the π π nearest vertex of the triangle is seen to be both r (csc − 1) and t (csc + 1), so 8 8 π π π 2 2 r (csc − 1) (1 + sin ) (1 − sin ) π 2 8 8 8 rt = = = 1 − sin π π csc + 1 1 + sin 8 8 8 √ √ 2 π π 1 − 2 sin cos 1 1 1 2 2 + 2 8 4 = + = + = + = . 2 2 2 2 2 4 4