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Geometry Collinearity

Source: 2015 Indonesia Math Olympiad Day 1 Problem 3

June 30, 2017

geometrycircumcircle

Problem Statement

Given an acute triangle

ABC

.

\Gamma _{B}

is a circle that passes through

AB

, tangent to

AC

at

A

and centered at

O_{B}

. Define

\Gamma_C

and

O_C

the same way. Let the altitudes of

\triangle ABC

from

B

and

C

meets the circumcircle of

\triangle ABC

at

X

and

Y

, respectively. Prove that

A

, the midpoint of

XY

and the midpoint of

O_{B}O_{C}

is collinear.

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