HMMT 二月 2013 · 冲刺赛 · 第 19 题

英文解析

19. [ 11 ] An isosceles trapezoid ABCD with bases AB and CD has AB = 13, CD = 17, and height 3. Let
    E be the intersection of AC and BD . Circles Ω and ω are circumscribed about triangles ABE and
    CDE . Compute the sum of the radii of Ω and ω .
    Answer: 39 Let Ω have center O and radius R and let ω have center P and radius M . Let Qbe the intersection of AB and OE . Note that OE is the perpendicular bisector of AB because thetrapezoid is isosceles. Also, we see OE is the circumradius of Ω. On the other hand, we know by
    13 13
    similarity of △ AEB and △ CED that QE = · 3 = · 3. And, because BQ = 13 / 2 and is
    13+17 30
    √
    ( )
    2132
    perpendicular to OQ , we can apply the Pythagorean theorem to △ OQB to see OQ = R − .
    √2
    ( )
    13 13 132
    Since OE = OQ + QE , R = · 3 + R − . Solving this equation for R yields R = · 39.2
    30 2 30
    17 30
    Since by similarity M = R , we know R + M = R , so R + M = 39.
    13 13
    3 2