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As l varies, circumcircle of XYZ passes through a fixed point

Source: 2023 Turkey Egmo Tst P6

March 23, 2023

geometrycircumcircle

Problem Statement

Let

ABC

be a scalene triangle and

l_0

be a line that is tangent to the circumcircle of

ABC

at point

A

. Let

l

be a variable line which is parallel to line

l_0

. Let

l

intersect segment

AB

and

AC

at the point

X

,

Y

respectively.

BY

and

CX

intersects at point

T

and the line

AT

intersects the circumcirle of

ABC

at

Z

. Prove that as

l

varies, circumcircle of

XYZ

passes through a fixed point.

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