Equal Areas of Vertex Triangles in a Regular Polygon

We determine the number of distinct areas of triangles formed by three vertices of a regular n-gon. Equal-area pairs are classified by reducing the problem to conjugation-stable vanishing relations of twelve roots of unity and applying the classification of Poonen and Rubinstein together with a short computer-assisted exact case analysis. This yields a closed formula for the number of distinct areas. The accompanying programs reproduce the finite classification step and independently verify the formula.