PUMaC 2008 · 几何（B 组） · 第 6 题

PUMaC 2008 — Geometry (Division B) — Problem 6

(4 points) Circles A , B , and C each have radius r , and their centers are the vertices of an equilateral triangle of side length 6 r . Two lines are drawn, one tangent to A and C and one tangent to B and C , such that A is on the opposite side of each line from B and C . Find the sine of the angle between the two lines. A C B 1 1 Geometry 2 2

解析

Two externally tangent circles have radius 2 and radius 3. Two lines are drawn, each tangent to both circles, but not at the point where the circles are tangent to each other. What is the area of the quadrilateral whose vertices are the four points of tangency between the circles and the lines? √ 48 6 ( ANS: 5 Let O and P be the centers of the circles of radius 2 and radius 3, respectively, and let A and B be the corresponding points of tangency with the line, so that OA = 2, P B = 3, and OP = 5. If C is the point on BP that is 1 unit away from P , then AOCB is a rectangle, so ∠ OCP is a right √ √ ′ ′ 2 2 angle, so CO = 5 − 1 = 2 6. If B is the projection of B onto OP , then BP B is similar to √ √ P B 3 3 P B 3 6 6 ′ ′ ′ OP C , so P B = P C = · 1 = and B B = CO = · 2 6 = . Similarly, if A is the 5 5 5 5 OP OP √ 2 4 6 ′ ′ projection of A onto OP , then OA = and A A = . So 5 5 2 3 24 ′ ′ ′ ′ A B = OP + OA − P B = 5 + − = . So, the area of the quadrilateral, twice the area of 5 5 5 ( ) √ √ √ 24 1 6 6 4 6 48 6 ′ ′ ABB A , is 2 · · + = CB: IAF) 5 2 5 5 5