PUMaC 2016 · 几何（A 组） · 第 1 题
PUMaC 2016 — Geometry (Division A) — Problem 1
题目详情
- Let 4 ABC be an equilateral triangle with side length 1 and let Γ the circle tangent to AB and AC at B and C , respectively. Let P be on side AB and Q be on side AC so that P Q ‖ BC , and the circle through A , P , and Q is tangent to Γ. If the area of 4 AP Q can be written in √ a the form for positive integers a and b , where a is not divisible by the square of any prime, b find a + b .
解析
- Let T be the point of tangency and D be the intersection of AT and BC ; note that AD is an ◦ 1 AD √ altitude. Since ∠ BT C = 120 , we find that T D = = . Let K be the intersection of 3 2 3 ( ) ◦ AK 3 1 AD AT and P Q ; since ∠ AP K = ∠ P T K = 60 , = 3. Then, AK = · 1 − · AD = , so KT 1+3 3 2 √ 3 P, Q are the midpoints of AB, AC , respectively. The area of 4 AP Q is then , so a + b = 19 . 16 Problem written by Mel Shu and Bill Huang.