HMMT 二月 2012 · 几何 · 第 4 题
HMMT February 2012 — Geometry — Problem 4
题目详情
英文原题
There are circles ω1andω2. They intersect in two points, one of which is the point A.Blies on ω1
such that ABis tangent to ω2. The tangent to ω1atBintersects ω2atCandD, where Dis the closer
toB.AD intersects ω1again at E. IfBD = 3 and CD = 13, find EB/ED .
解析
英文解析
- There are circles ω and ω . They intersect in two points, one of which is the point A . B lies on ω
1 2 1
such that AB is tangent to ω . The tangent to ω at B intersects ω at C and D , where D is the closer
2 1 2
to B . AD intersects ω again at E . If BD = 3 and CD = 13, find EB/ED .
√1
Answer: 4 3 / 3
[diagram]
√
√ √
By power of a point, BA = BD · BC = 4 3. Also, DEB ∼ DBA , so EB/ED = BA/BD = 4 3 / 3.