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Two internal tangent circles

Source: Ukrainian TST 2008 Problem 4

February 12, 2009
geometrygeometry unsolved

Problem Statement

Two circles ω1 \omega_1 and ω2 \omega_2 tangents internally in point P P. On their common tangent points A A, B B are chosen such that P P lies between A A and B B. Let C C and D D be the intersection points of tangent from A A to ω1 \omega_1, tangent from B B to ω2 \omega_2 and tangent from A A to ω2 \omega_2, tangent from B B to ω1 \omega_1, respectively. Prove that CA \plus{} CB \equal{} DA \plus{} DB.
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