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3 equal incircles wanted

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2018 Shortlist G1

October 2, 2021
geometryequal segmentsequal circles

Problem Statement

Let OO be the center of the circle circumscribed to △ABC\vartriangle ABC, and let P P be any point on BCBC (P≠BP \ne B and P≠CP \ne C). Suppose that the circle circumscribed to △BPO\vartriangle BPO intersects ABAB at RR (R≠AR \ne A and R≠BR \ne B) and that the circle circumscribed to △COP\vartriangle COP intersects CACA at point QQ (Q≠CQ \ne C and Q≠AQ \ne A). 1) Show that △PQR∼△ABC\vartriangle PQR \sim \vartriangle ABC and thatO O is orthocenter of △PQR\vartriangle PQR. 2) Show that the circles circumscribed to the triangles △BPO\vartriangle BPO, △COP\vartriangle COP, and △PQR\vartriangle PQR all have the same radius.
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