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A bisector in a complete cyclic quadrilateral

Source: 4th Memorial Mathematical Competition "Aleksandar Blazhevski - Cane"- Senior D1 P3

December 21, 2022

geometrycyclic quadrilateralcomplete quadrilateraltangent circlesangle bisector

Problem Statement

Let

ABCD

be a cyclic quadrilateral inscribed in a circle

\omega

with center

O

. The lines

AD

and

BC

meet at

E

, while the lines

AB

and

CD

meet at

F

. Let

P

be a point on the segment

EF

such that

OP \perp EF

. The circle

\Gamma_{1}

passes through

A

and

E

and is tangent to

\omega

at

A

, while

\Gamma_{2}

passes through

C

and

F

and is tangent to

\omega

at

\angle XPY

. If

\Gamma_{1}

and

\Gamma_{2}

meet at

X

and

Y

, prove that

PO

is the bisector of

\angle XPY

.
Proposed by Nikola Velov

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