HMMT 二月 2008 · 几何 · 第 7 题

英文解析

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 .
   Answer: 8 Let D be the point of tangency between C and C . We see that T is the radical center 3
   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.
   1CA
   3CDT
   2C
   We have b
   ∠ 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 Dtan + 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