Rational Area Profiles of Regular Polygons Cut by Two Congruent Diagonals

Source abstract

Two congruent diagonals of a regular hexagon cut it into four regions whose areas are all rational multiples of the area of the hexagon. We determine when this phenomenon occurs in a regular $n$-gon. For every regular polygon cut by two intersecting congruent diagonals, neither of which is a diameter, we classify all unions of the four resulting regions whose areas are rational fractions of the total area. The hexagon is the only configuration in which all four regions are simultaneously rational. One of the two regions on the axis of symmetry is rational in exactly five sporadic configurations, with $n=6,18,30$, whereas the other region and the adjacent two-region unions belong to infinite families with simple geometric explanations. The two lateral regions have rational area ratios in exactly two configurations, with $n=6$ and $n=12$. The proof reduces each area ratio to a trigonometric quotient, bounds its rational values by norms in cyclotomic fields, and settles the remaining equations using Myerson's classification of equal products of two sines and the Poonen--Rubinstein classification of a three-sine-product equation.