MathDB

It is known that circle

\Gamma_1(O_1)

has center at

O_1

, circle

\Gamma_2(O_2)

has center at

O_2

, and both intersect at points

C

and

D

. It is also known that points

P

and

Q

lie on circles

\Gamma_1(O_1)

and

\Gamma_2(O_2)

, respectively. ). A line

\ell

passes through point

D

and intersects

\Gamma_1(O_1)

and

\Gamma_2(O_2)

at points

A

and

B

, respectively. The lines

PD

and

AC

meet at point

M

, and the lines

QD

and

BC

meet at point

N

. Let

O

be center outer circle of triangle

ABC

. Prove that

OD

is perpendicular to

MN

if and only if a circle can be found which passes through the points

P, Q, M

and

N

Problem Statement