Tuesday, November 29th

Problem
Find all the possible areas on an hexagon with equal angles and whose sides are 1,2,3,4,5, and 6 in some order.

Solution
Up to cyclic permutations and reflexions, the only possibilities for the sides are 1,6,2,4,3,5 and 1,6,3,2,5,4. Thus the possible values of areas are $\frac{67\sqrt{3}}{4}$ and $\frac{65\sqrt{3}}{4}$.