MathDB

G1

Part of 2018 Costa Rica - Final Round

Problems(1)

3 equal incircles wanted

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

10/2/2021
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.
geometryequal segmentsequal circles