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A cyclic quadrilateral

Source: Bulgarian NMO 2017, 3rd round, p.1

April 22, 2017

geometry

Problem Statement

An convex qudrilateral

ABCD

is given.

O

is the intersection point of the diagonals

AC

and

BD

. The points

A_1,B_1,C_1, D_1

lie respectively on

AO, BO, CO, DO

such that

AA_1=CC_1, BB_1=DD_1

. The circumcircles of

\triangle AOB

and

\triangle COD

meet at second time at

M

and the the circumcircles of

\triangle AOD

and

\triangle BOC

- at

N

. The circumcircles of

\triangle A_1OB_1

and

\triangle C_1OD_1

meet at second time at

P

and the the circumcircles of

\triangle A_1OD_1

and

\triangle B_1OC_1

- at

Q

. Prove that the quadrilateral

MNPQ

is cyclic.

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