Question 159
A convex hexagon is given in which any two opposite sides have the following property: the distance between their midpoints is $\frac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.
(A convex hexagon $ABCDEF$ has three pairs of opposite sides: $AB$ and $DE$, $BC$ and $EF$, $CD$ and $FA$.)
(A convex hexagon $ABCDEF$ has three pairs of opposite sides: $AB$ and $DE$, $BC$ and $EF$, $CD$ and $FA$.)