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In Euclidean geometry, a regular polygon is equiangular (all angles are equal in size) and equilateral (all sides have the same length) polygon. So regular polygons should be thought of as special polygons. [ABSTRACT FROM AUTHOR]

Based on Peter's work from 2003, quadrilaterals can be characterized in the following way: "among all quadrilaterals with given side lengths , , and , those of the largest possible area are exactly the cyclic ones". In this paper, we will give the corresponding characterization for every polygon, by means of quasicyclic polygons properties. [ABSTRACT FROM AUTHOR]

A Pythagorean triple is an ordered triple of integers (a,b,c) = (0,0,0) such that a² +b² = c². It is well known that the set P of all Pythagorean triples has an intrinsic structure of commutative monoid with respect to a suitable binary operation, (P,*). In this article, we will introduce the "commensurability" relation R among Pythagorean triples, and we will see that it induces a group quotient, P/R, which is isomorphic with the direct product of infinite (countable) copies of C∞, the infinite cyclic group, and a cyclic group of order 4. As an application, we will see that the acute angles of Pythagorean triangles are irrational when measured in degrees. [ABSTRACT FROM AUTHOR]