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2026 AMC 8 Problem 25: Equiangular Hexagons in Triangle

In an equiangular hexagon, all interior angles measure 120°. An example of such a hexagon with side lengths 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 △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.