Let ABCD be a convex quadrilateral, and let M and N be the midpoints of sides AD and BC, respectively. Prove that MN<=(AB+CD)/2. When does equality occur?
Equality occurs when the quadrilateral is a square. In a square, all the sides are equal, so the midpoints are at the same height and the segment connecting them is the same length as any side of the square. Equality also occurs within a trapezoid, because the midline of any trapezoid is half the sum of the bases.