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common tangent of externally tangent circles wanted

Source: Slovenia TST 1997 p1 - Swiss TST 2005 p4

February 15, 2020
common tangenttangenttangent circlesgeometry

Problem Statement

Circles K1K_1 and K2K_2 are externally tangent to each other at AA and are internally tangent to a circle KK at A1A_1 and A2A_2 respectively. The common tangent to K1K_1 and K2K_2 at AA meets KK at point PP. Line PA1PA_1 meets K1K_1 again at B1B_1 and PA2PA_2 meets K2K_2 again at B2B_2. Show that B1B2B_1B_2 is a common tangent of K1K_1 and K2K_2.
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