MathDB

OP circles

Source: EGMO 2022/6

April 9, 2022

EGMOgeometrycirclesEGMO2022

Problem Statement

Let

ABCD

be a cyclic quadrilateral with circumcenter

O

. Let the internal angle bisectors at

A

and

B

meet at

X

, the internal angle bisectors at

B

and

C

meet at

Y

, the internal angle bisectors at

C

and

D

meet at

Z

, and the internal angle bisectors at

D

and

A

meet at

W

. Further, let

AC

and

BD

meet at

P

. Suppose that the points

X

,

Y

,

Z

,

W

,

O

, and

P

are distinct. Prove that

O

,

X

,

Y

,

Z

,

W

lie on the same circle if and only if

P

,

X

,

Y

,

Z

, and

W

lie on the same circle.

Back to Problems View on AoPS