Problem

Source: Junior Olympiad of Malaysia 2013 P5

Tags: geometry



Consider a triangle ABC with height AH and H on BC. Let γ1 and γ2 be the circles with diameter BH,CH respectively, and let their centers be O1 and O2. Points X,Y lie on γ1,γ2 respectively such that AX,AY are tangent to each circle and X,Y,H are all distinct. P is a point such that PO1 is perpendicular to BX and PO2 is perpendicular to CY. Prove that the circumcircles of PXY and AO1O2 are tangent to each other.