HMMT 二月 2007 · 几何 · 第 3 题
HMMT February 2007 — Geometry — Problem 3
题目详情
英文原题
- [ 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
英文解析
- [ 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 Rmidpoints of segments P R and P Q , so [ P QR ] = · [ P BA ] = 2 · 2[ P BA ] = 2 3.
P A P B