HMMT 二月 2014 · 几何 · 第 9 题

英文解析

9. Two circles are said to be orthogonal if they intersect in two points, and their tangents at either point ofintersection are perpendicular. Two circles ω and ω with radii 10 and 13, respectively, are externally
   1 2
   √
   tangent at point P . Another circle ω with radius 2 2 passes through P and is orthogonal to both ω
   3 1
   and ω . A fourth circle ω , orthogonal to ω , is externally tangent to ω and ω . Compute the radius
   2 4 3 1 2
   of ω .
   Answer: Let ω have center O and radius r . Since ω is orthogonal to ω , ω , ω , it has equal 924
   i i i 3 1 2 4
   power r to each of them. Thus O is the radical center of ω , ω , ω , which is equidistant to the three 261
   3 1 2 4
   sides of △ O O O and therefore its incenter.3
   1 2 4
   For distinct i, j ∈ { 1 , 2 , 4 } , ω ∩ ω lies on the circles with diameters O O and O O , and hence ωi j 3 i 3 j 3
   r r r 130 r 92
   1 2 4 42
   itself. It follows that ω is the incircle of △ O O O , so 8 = r = = = ⇒ r = .
   3 1 2 4 4
   r + r + r 23+ r 613
   1 2 4 4
   Comment: The condition P ∈ ω is unnecessary. 3