Question 25
In an *equiangular hexagon*, all interior angles measure $120^\circ.$ An example of such a hexagon with side lengths of $2,$ $3,$ $1,$ $3,$ $2,$ and $2$ is shown below, inscribed in equilateral triangle $ABC.$ Consider all equiangular hexagons with positive integer side lengths that can be inscribed in $\triangle ABC,$ with all six vertices on the sides of the triangle. What is the total number of such hexagons? Hexagons that differ only by a rotation or a reflection are considered the same.
