A Curious Property of Octagons

The famous theorem of Van Aubel for quadrilaterals postulates that if squares are built externally on the sides of any quadrilateral, then the two segments that join the opposing centers of these squares are congruent and orthogonal. Inspired by this result and also by the results of Krishna, in this article we will prove the following result of plane geometry: each octagon is associated with a parallelogram, in some cases the parallelogram in question can be degenerate at a point or a segment. This is possible because of complex numbers and basics of analytical geometry.