Achilleas Sinefakopoulos, M. N. Raptou

Dec 1, 2015

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Mathematics Magazine

Abstract

BA2 A2C · CB1 B1 A · AC1 C1 B = 1. From these two relations it readily follows that BA1 : A1C = BA2 : A2C . Denote this common ratio by r1 and note that the points A1 and A2 lie on the Apollonian circle ω1 of diameter A1 A2, which is the locus of points P in the plane such that BP : PC = r1. Working similarly we see that the points B1 and B2 lie on the Apollonian circle ω2 of diameter B1 B2, which is the locus of points P in the plane such that CP : PA = CB1 : B1 A = CB2 : B2 A := r2, while the points C1 and C2 lie on the Apollonian circle ω3 of diameter C1C2, which is the locus of points P in the plane such that

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