MathDB

Let

ABCD

be a convex quadrilateral whose sides

AD

and

BC

are not parallel. Suppose that the circles with diameters

AB

and

CD

meet at points

E

and

F

inside the quadrilateral. Let

\omega_E

be the circle through the feet of the perpendiculars from

E

to the lines

AB,BC

and

CD

. Let

\omega_F

be the circle through the feet of the perpendiculars from

F

to the lines

CD,DA

and

AB

. Prove that the midpoint of the segment

EF

lies on the line through the two intersections of

\omega_E

and

\omega_F

.
Proposed by Carlos Yuzo Shine, Brazil

Problem Statement