HMMT 二月 2006 · TEAM2 赛 · 第 13 题

HMMT February 2006 — TEAM2 Round — Problem 13

[25] Four circles with radii 1 , 2 , 3 , and r are externally tangent to one another. Compute r . (No proof is necessary.)

解析

[25] Four circles with radii 1 , 2 , 3 , and r are externally tangent to one another. Compute r . (No proof is necessary.) Answer: 6 / 23 Solution: Let A, B, C, P be the centers of the circles with radii 1, 2, 3, and r , respectively. Then, ABC is a 3-4-5 right triangle. Using the law of cosines in 4 P AB yields 2 2 2 3 + (1 + r ) − (2 + r ) 3 − r cos ∠ P AB = = 2 · 3 · (1 + r ) 3(1 + r ) Similarly, 2 2 2 4 + (1 + r ) − (3 + r ) 2 − r cos ∠ P AC = = 2 · 4 · (1 + r ) 2(1 + r ) 2 2 We can now use the equation (cos ∠ P AB ) + (cos ∠ P AC ) = 1 , which yields 0 = 2 23 r + 132 r − 36 = (23 r − 6)( r + 6), or r = 6 / 23 .