HMMT 二月 2006 · 几何 · 第 7 题

HMMT February 2006 — Geometry — Problem 7

Suppose ABCD is an isosceles trapezoid in which AB ‖ CD . Two mutually externally tangent 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, 2 compute the radius of either circle.

解析

Suppose ABCD is an isosceles trapezoid in which AB ‖ CD . Two mutually externally tangent 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, 2 compute the radius of either circle. 3 Answer: 7 Solution: Let the radius of both circles be r , and let ω be centered at O . Let ω be 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 2 1 2 7 3 Since O Q ⊥ BC , r = O Q = BQ · QC = ( − r )(3 − r ) = r − r + . Solving, we 1 1 2 2 2 3 find r = . 7