MathDB

Let the sides

AD

and

BC

of the quadrilateral

ABCD

(such that

AB

is not parallel to

CD

) intersect at point

P

. Points

O_1

and

O_2

are circumcenters and points

H_1

and

H_2

are orthocenters of triangles

ABP

and

CDP

, respectively. Denote the midpoints of segments

O_1H_1

and

O_2H_2

E_1

and

E_2

, respectively. Prove that the perpendicular from

E_1

CD

, the perpendicular from

E_2

AB

and the lines

H_1H_2

are concurrent.
Proposed by Eugene Bilopitov, Ukraine

Problem Statement