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 an elementary phase reduction. This yields a complete classification of the possible equal-area coincidences and a closed formula for the number of distinct areas. The proof is entirely computer-free.