MathDB

5.

Part of JOM 2013

Problems(1)

Tangent circles

Source: Junior Olympiad of Malaysia 2013 P5

7/20/2015
Consider a triangle ABCABC with height AHAH and HH on BCBC. Let γ1\gamma_1 and γ2\gamma_2 be the circles with diameter BH,CHBH,CH respectively, and let their centers be O1O_1 and O2O_2. Points X,YX,Y lie on γ1,γ2\gamma_1,\gamma_2 respectively such that AX,AYAX,AY are tangent to each circle and X,Y,HX,Y,H are all distinct. PP is a point such that PO1PO_1 is perpendicular to BXBX and PO2PO_2 is perpendicular to CYCY.
Prove that the circumcircles of PXYPXY and AO1O2AO_1O_2 are tangent to each other.
geometry