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Perpendiculars dropped from circumcircle's tangent

Source: Problem 6 of Russian Regional Olympiad 2010 Grade 9

September 25, 2011
geometrycircumcirclegeometry proposed

Problem Statement

Let points AA, BB, CC lie on a circle, and line bb be the tangent to the circle at point BB. Perpendiculars PA1PA_1 and PC1PC_1 are dropped from a point PP on line bb onto lines ABAB and BCBC respectively. Points A1A_1 and C1C_1 lie inside line segments ABAB and BCBC respectively. Prove that A1C1A_1C_1 is perpendicular to ACAC.
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