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Circumcircle excircle chaos

Source: ISL 2021 G8

July 12, 2022

geometrycircumcircleexcircleDDITISL 2021

Problem Statement

Let

ABC

be a triangle with circumcircle

\omega

and let

\Omega_A

be the

A

-excircle. Let

X

and

Y

be the intersection points of

\omega

and

\Omega_A

. Let

P

and

Q

be the projections of

A

onto the tangent lines to

\Omega_A

at

X

and

Y

respectively. The tangent line at

P

to the circumcircle of the triangle

APX

intersects the tangent line at

Q

to the circumcircle of the triangle

AQY

at a point

R

. Prove that

\overline{AR} \perp \overline{BC}

.

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