Association of two square difference identity to regular polygons and circles

Abstract

GeoGebra is a dynamic software that is frequently used and of increasing importance in mathematics teaching processes in our digital age. Accordingly, in this study a new perspective has been brought to the proofs of the “two square difference identity” expressed for the square, which is a flat polygon, made with different approaches. With side lengths a, b, and a>b, it has been shown that the identity given by the equation (difference of area) a²-b²=(a-b)(a+b) is true for other regular polygons as well. In the study, direct proof method was used within the framework of the principle of conservation of measure, which is one of the basic principles of geometry teaching. GeoGebra program, which is a dynamic geometry software, was preferred for drawing geometric shapes used in proofs. In order to generalize the number n, a different fragmentation technique was preferred to the proofs made using different drawings for equilateral triangle and square, which are the simplest regular polygons. It has also been shown that this identity is true for circles viewed as polygons with an infinite number of sides.