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Iran TST P4

Source: Iranian TST 2022 problem 4

April 2, 2022

geometry

Problem Statement

Cyclic quadrilateral

ABCD

with circumcenter

O

is given. Point

P

is the intersection of diagonals

AC

and

BD

. Let

M

and

N

be the midpoint of the sides

AD

and

BC

, respectively. Suppose that

BCP

,

\omega_2

and

\omega_3

be the circumcircle of triangles

ADP

,

BCP

and

OMN

, respectively. The intersection point of

E

and

\omega_3

, which is not on the arc

APD

of

\omega_1

, is

E

and the intersection point of

\omega_2

and

\omega_3

, which is not on the arc

BPC

of

\omega_2

, is

F

. Prove that

OF=OE

.
Proposed by Seyed Amirparsa Hosseini Nayeri

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