HMMT 十一月 2010 · 团队赛 · 第 5 题

英文解析

5. [ 4 ] Circle O has chord AB . A circle is tangent to O at T and tangent to AB at X such that AX = 2 XB .
   What is ?AT
   Answer: 2 Let T X meet circle O again at Y . Since the homethety centered at T takes X to YBTalso takes AB to the tangent line of circle O passing through Y , we have Y is the midpoint of arc AB .
   AT AX
   This means that ∠ AT Y = ∠ Y T B . By the Angle Bisector Theorem, = = 2.
   BT BX