Convex quadrilateral ABCD is given. Lines BC and AD intersect at point O, with B lying on the segment OC, and A on the segment OD. I is the center of the circle inscribed in the OAB triangle, J is the center of the circle exscribed in the triangle OCD touching the side of CD and the extensions of the other two sides. The perpendicular from the midpoint of the segment IJ on the lines BC and AD intersect the corresponding sides of the quadrilateral (not the extension) at points X and Y. Prove that the segment XY divides the perimeter of the quadrilateralABCD in half, and from all segments with this property and ends on BC and AD, segment XY has the smallest length. geometryincircleexcircleperimeterbisectsmidpointminimum