HMMT 二月 2007 · 几何 · 第 3 题

HMMT February 2007 — Geometry — Problem 3

[ 4 ] Circles ω , ω , and ω are centered at M, N, and O , respectively. The points of tangency between 1 2 3 ω and ω , ω and ω , and ω and ω are tangent at A , B , and C , respectively. Line M O intersects 2 3 3 1 1 2 ω and ω again at P and Q respectively, and line AP intersects ω again at R . Given that ABC is 3 1 2 an equilateral triangle of side length 1, compute the area of P QR .

解析

[ 4 ] Circles ω , ω , and ω are centered at M, N, and O , respectively. The points of tangency between 1 2 3 ω and ω , ω and ω , and ω and ω are tangent at A , B , and C , respectively. Line M O intersects 2 3 3 1 1 2 ω and ω again at P and Q respectively, and line AP intersects ω again at R . Given that ABC is 3 1 2 an equilateral triangle of side length 1, compute the area of P QR . √ Answer: 2 3 . Note that ON M is an equilateral triangle of side length 2, so m ∠ BP A = m ∠ BOA/ 2 = √ π/ 6. Now BP A is a 30-60-90 triangle with short side length 1, so AP = 3. Now A and B are the √ P Q P R midpoints of segments P R and P Q , so [ P QR ] = · [ P BA ] = 2 · 2[ P BA ] = 2 3. P A P B