Equal Areas of Vertex Triangles in a Regular Polygon

For a regular n-gon, let a(n) be the number of distinct areas of triangles formed by its vertices. We classify all coincidences between the areas of noncongruent vertex triangles. Equality of areas is reduced to a conjugation-stable vanishing relation of twelve roots of unity with additional phase constraints. Together with the classification of Poonen and Rubinstein, these constraints leave one rational one-parameter family, one sporadic pattern of denominator 24, and two of denominator 30. This yields an explicit formula for a(n), exact multiplicity results, a rational generating function, quadratic quasipolynomiality with minimal quasiperiod 120, and recurrences for the counting sequence. No area is attained by more than two noncongruent triangle shapes. The classification proof is entirely computer-free. Version 3. All finite cases in the phase classification are now treated explicitly, making the classification proof entirely computer-free. Further consequences of the classification have been added, including the exact multiplicity of area values, a rational generating function, quadratic quasipolynomiality with minimal quasiperiod 120, and recurrences for the counting sequence. The associated integer sequence is OEIS A400213.