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prove that the circumcenter of a triangle moves along a straight line

Source: Sharygin Geometry Olympiad 2014 Correspondence Round P13

August 1, 2018
geometryLocusCircumcenter

Problem Statement

Let ACAC be a fixed chord of a circle ω\omega with center OO. Point BB moves along the arc ACAC. A fixed point PP lies on ACAC. The line passing through PP and parallel to AOAO meets BABA at point A1A_1, the line passing through PP and parallel to COCO meets BCBC at point C1C_1. Prove that the circumcenter of triangle A1BC1A_1BC_1 moves along a straight line.
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