HMMT 二月 2008 · 几何 · 第 7 题

7. [ 6 ] Let C and C be externally tangent circles with radius 2 and 3, respectively. Let C be a circle 1 2 3 internally tangent to both C and C at points A and B , respectively. The tangents to C at A and B 1 2 3 meet at T , and T A = 4. Determine the radius of C . 3 Answer: 8 Let D be the point of tangency between C and C . We see that T is the radical center 1 2 of the three circles, and so it must lie on the radical axis of C and C , which happens to be their 1 2 common tangent T D . So T D = 4. A C 1 T D C 3 C 2 B We have ∠ AT D 2 1 ∠ BT D 3 3 tan = = , and tan = = . 2 T D 2 2 T D 4 Thus, the radius of C equals to 3 ( ) ∠ AT B ∠ AT D + ∠ BT D T A tan = 4 tan 2 2 ∠ AT D ∠ BT D tan + tan 2 2 = 4 · ∠ AT D ∠ BT D 1 − tan tan 2 2 1 3

2 4 = 4 · 1 3 1 − · 2 4 = 8 . 1