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A,M,X are collinear

Source: JBMO 2017, Q3

June 26, 2017

geometry

Problem Statement

Let

ABC

be an acute triangle such that

AB\neq AC

,with circumcircle

\Gamma

and circumcenter

O

. Let

M

be the midpoint of

BC

and

D

be a point on

\Gamma

such that

AD \perp BC

. let

T

be a point such that

BDCT

is a parallelogram and

Q

a point on the same side of

BC

as

A

such that

\Gamma

and

\angle{CQM}=\angle{CBA}

. Let the line

AO

intersect

\Gamma

at

E

(E\neq A)

and let the circumcircle of

\triangle ETQ

intersect

\Gamma

at point

X\neq E

. Prove that the point

A,M

and

X

are collinear.

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