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Quadratic Forms, Chebyshev Polynomials, and Geometric Inequalities

Abstract We discuss a geometric inequality for 2n-gons proved in [Mushkarov, O., Nikolov N. (2006). Semiregular polygons. Amer. Math. Monthly. 113(4): 339–344] and show how the algebraic inequality it is based on can be proved by using the standard theory of quadratic forms. In addition, we prove an “odd” version of the geometric inequality which leads to some interesting geometric problems for polygons with an odd number of sides.