HMMT 二月 2001 · 冲刺赛 · 第 10 题

HMMT February 2001 — Guts Round — Problem 10

[8] Two concentric circles have radii r and R > r . Three new circles are drawn so that they are each tangent to the big two circles and tangent to the other two new circles. R Find . r

解析

[8] Two concentric circles have radii r and R > r . Three new circles are drawn so that they are each tangent to the big two circles and tangent to the other two new circles. R Find . r Solution: The centers of the three new circles form a triangle. The diameter of the new circles is R − r , so the side length of the triangle is R − r . Call the center of the concentric circle O , two vertices of the triangle A and B , and AB ’s midpoint D . OA is the average R sin(30) sin(90) R + r and r , namely . Using the law of sines on triangle DAO , we get = ⇒ R = 3 r , 2 AD AO R so = 3 . r