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Tangent circles

Source: Junior Olympiad of Malaysia 2013 P5

July 20, 2015

geometry

Problem Statement

Consider a triangle

ABC

with height

AH

and

H

on

BC

. Let

\gamma_1

and

\gamma_2

be the circles with diameter

\gamma_1,\gamma_2

respectively, and let their centers be

O_1

and

O_2

. Points

X,Y

lie on

\gamma_1,\gamma_2

respectively such that

AX,AY

are tangent to each circle and

X,Y,H

are all distinct.

P

is a point such that

PO_1

is perpendicular to

BX

and

PO_2

is perpendicular to

CY

.
Prove that the circumcircles of

PXY

and

AO_1O_2

are tangent to each other.

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