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junior collinearity, intersecting circles related

Source: 2022 Moldova JBMO TST p3

November 4, 2022
geometrycollinear

Problem Statement

Circles ω1\omega_1 and ω2\omega_2 intersect at points AA and BB. A straight line is drawn through point BB, which again intersects circles ω1\omega_1 and ω2\omega_2 at points CC and DD, respectively. Point EE, located on circle ω1\omega_1 , satisfies the relation CE=CBCE = CB , and point FF, located on circle ω2\omega_2, satisfies the relation DB=DFDB = DF. The line BFBF intersects again the circle ω1\omega_1 at the point PP, and the line BEBE intersects again the circle ω2\omega_2 at the point QQ. Prove that the points A,PA, P, and QQ are collinear.
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