Quadratic identities of polygons (Q1364857)

If \(ABCD\) is a quadrilateral and \(M,N\) are the mid-points of the diagonals, then \(|AB |^2+ |BC |^2+ |CD |^2+ |DA|^2 =|AC|^2 + |BD |^2 +4|MN |^2\). This theorem has given L. Euler in 1748. (His proof is a good and simple exercise in analytic geometry.) In 1981 \textit{A. J. Douglas} carried a generalization of this Euler theorem on polygons with even number of points [Math. Gaz. 65, 19-22 (1981; Zbl 0452.51023)]. For the Fourier analysis of polygons the author develops analytic instruments [Math. Semesterber. 41, No. 1, 23-42 (1994; Zbl 0798.51024)]. On this basis the author establishes a wide class of quadratic identities of polygons, whose geometrical interpretation leads to polygon theorems. In particular a quadratic identity for the tetrahedron \(ABCD\) is \[ |AD |^2+ |BD |^2 |+ |CD |^2=3 |SD |^2+ 1/3\bigl(|AB |^2+ |BC|^2|+|CA |^2), \] where \(D\) is the intersection point of the medians of the angles \(ABC\). In this paper geometrical transformations of polygons are used and referring to this the quadratic identities.