MathDB

In a convex quadrilateral

ABCD

, the lines

AB

and

CD

meet at point

E

and the lines

AD

and

BC

meet at point

F

. Let

P

be the intersection point of diagonals

AC

and

BD

. Suppose that

\omega_1

is a circle passing through

D

and tangent to

AC

P

. Also suppose that

\omega_2

is a circle passing through

C

and tangent to

BD

P

. Let

X

be the intersection point of

\omega_1

and

AD

, and

Y

be the intersection point of

\omega_2

and

BC

. Suppose that the circles

\omega_1

and

\omega_2

intersect each other in

Q

for the second time. Prove that the perpendicular from

P

to the line

EF

passes through the circumcenter of triangle

XQY

. Proposed by Iman Maghsoudi

Problem Statement