PUMaC 2023 · 几何（B 组） · 第 3 题

PUMaC 2023 — Geometry (Division B) — Problem 3

Define a common chord between two intersecting circles to be the line segment connecting their two intersection points. Let ω , ω , ω be three circles of radii 3, 5, and 7, respectively. 1 2 3 Suppose they are arranged in such a way that the common chord of ω and ω is a diameter 1 2 of ω , the common chord of ω and ω is a diameter of ω , and the common chord of ω and 1 1 3 1 2 ω is a diameter of ω . Compute the square of the area of the triangle formed by the centers 3 2 of the three circles. √

解析

Define a common chord between two intersecting circles to be the line segment connecting their two intersection points. Let ω , ω , ω be three circles of radii 3, 5, and 7, respectively. 1 2 3 Suppose they are arranged in such a way that the common chord of ω and ω is a diameter 1 2 of ω , the common chord of ω and ω is a diameter of ω , and the common chord of ω and 1 1 3 1 2 ω is a diameter of ω . Compute the square of the area of the triangle formed by the centers 3 2 of the three circles. Proposed by Eric Shen Answer: 96 q 2 2 By Pythagoras, the distance between the centers of circles ω and ω with j > i is r − r . i j j i √ √ √ We seek the area of a triangle with sidelengths 16, 24, and 40. But this is a right triangle √ √ √ √ 1 2 whose area is · 16 · 24 = 4 6, and our answer is (4 6) = 96. 2 √