HMMT 十一月 2010 · 团队赛 · 第 5 题
HMMT November 2010 — Team Round — Problem 5
题目详情
- [ 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 . AT What is ? BT
解析
- [ 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 . AT What is ? BT Answer: 2 Let T X meet circle O again at Y . Since the homethety centered at T takes X to Y also 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