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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

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