RO-TST 2008 Problem 8

Show that each convex pentagon has a vertex from which the distance to the opposite side of the pentagon is strictly less than the sum of the distances from the two adjacent vertices to the same side.

Note. If the pentagon is labeled $ABCDE$ , the adjacent vertices of $A$ are $B$ and $E$ , the ones of $B$ are $A$ and $C$ etc.