IMO2003 problem A3

[eepiccolo]

Here is the third problem from the 44th International Mathematical Olympiad. The other five are posted also. I don't know the answers, but I'll be working on them when I get the chance.

A3. A convex hexagon has the property that for any pair of opposite sides the distance between their midpoints is (sqrt(3))/2 times the sum of their lengths. Show that all the hexagon's angles are equal.