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USAMO 2000 Problem 5

Source:

October 1, 2005
AMCUSA(J)MOUSAMOgeometrycircumcircleanalytic geometry

Problem Statement

Let A1A2A3A_1A_2A_3 be a triangle and let ω1\omega_1 be a circle in its plane passing through A1A_1 and A2.A_2. Suppose there exist circles ω2,ω3,…,ω7\omega_2, \omega_3, \dots, \omega_7 such that for k=2,3,…,7,k = 2, 3, \dots, 7, ωk\omega_k is externally tangent to ωk−1\omega_{k-1} and passes through AkA_k and Ak+1,A_{k+1}, where An+3=AnA_{n+3} = A_{n} for all n≥1n \ge 1. Prove that ω7=ω1.\omega_7 = \omega_1.
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