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externally tangent circles, prove collinearity of intersection points

Source: Mongolia MO 2000 Grade 10 P2

April 22, 2021
geometrycircles

Problem Statement

Circles ω1,ω2,ω3\omega_1,\omega_2,\omega_3 with centers O1,O2,O3O_1,O_2,O_3, respectively, are externally tangent to each other. The circle ω1\omega_1 touches ω2\omega_2 at P1P_1 and ω3\omega_3 at P2P_2. For any point AA on ω1\omega_1, A1A_1 denotes the point symmetric to AA with respect to O1O_1. Show that the intersection points of AP2AP_2 with ω3\omega_3, A1P3A_1P_3 with ω2\omega_2, and AP3AP_3 with A1P2A_1P_2 lie on a line.
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