Regular polygons on sides of special triangles

Abstract

For a natural number n ≥ 3 and a positive real number ξ let Pn (ξ) denote the area of the regular n-gon with side of length ξ and let P(ABC … Z) be the area of a convex polygon ABC … Z. In 1996 Josip Kovačević observed [1] that in every triangle ABC with sides a, b and c and with the angle A equal to π/3 radians the following relation holds: Similarly, if the angle A is equal to 2π/3 radians then On the other hand, when the angle A is π/2 radians, then Pythagoras’ theorem implies which could be rewritten as so because P(ABC) = bc/2. The last observation is the starting point for Veljan’s results in [2] and [3] that when the angle A is π - 2π/n radians and that when the angle A is 2π/n radians.