HMMT 二月 2006 · 几何 · 第 7 题
HMMT February 2006 — Geometry — Problem 7
题目详情
英文原题
- Suppose ABCD is an isosceles trapezoid in which AB ‖ CD . Two mutually externallytangent circles ω and ω are inscribed in ABCD such that ω is tangent to AB, BC ,
1 2 1
and CD while ω is tangent to AB, DA , and CD . Given that AB = 1 , CD = 6,
compute the radius of either circle.2
解析
英文解析
- Suppose ABCD is an isosceles trapezoid in which AB ‖ CD . Two mutually externallytangent circles ω and ω are inscribed in ABCD such that ω is tangent to AB, BC ,
1 2 1
and CD while ω is tangent to AB, DA , and CD . Given that AB = 1 , CD = 6,
compute the radius of either circle.2
Answer:3
Solution: Let the radius of both circles be r , and let ω be centered at O . Let ω be 7
1 1 1
tangent to AB, BC , and CD at P , Q , and R respectively. Then, by symmetry, P B =
1 1 − r and RC = 3 − r . By equal tangents from B and C , BQ = − r and QC = 3 − r .
2 2
°1
Now, ∠ BO C is right because m ∠ O BC + m ∠ BCO = ( m ∠ P BC + m ∠ BCR ) = 90 .
1 1 1
2 2 1 2 7 32
Since O Q ⊥ BC , r = O Q = BQ · QC = ( − r )(3 − r ) = r − r + . Solving, we
1 1
2 2 2
find r = .3 7