HMMT 二月 2013 · 冲刺赛 · 第 19 题

HMMT February 2013 — Guts Round — Problem 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 . 3 2

解析

[ 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 Q be the intersection of AB and OE . Note that OE is the perpendicular bisector of AB because the trapezoid 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 √ ( ) 2 13 2 perpendicular to OQ , we can apply the Pythagorean theorem to △ OQB to see OQ = R − . 2 √ ( ) 2 13 13 13 2 Since OE = OQ + QE , R = · 3 + R − . Solving this equation for R yields R = · 39. 30 2 30 17 30 Since by similarity M = R , we know R + M = R , so R + M = 39. 13 13 3 2