Varignon's and Wittenbauer's parallelograms

TL;DR

This paper introduces the concept of homothetic parallelograms as a generalization of Varignon's and Wittenbauer's parallelograms for arbitrary quadrangles, providing formulas for area and perimeter, and proving their similarity and perspectivity.

Contribution

It generalizes Varignon's and Wittenbauer's parallelograms to non-crossed quadrangles and derives formulas for their area and perimeter.

Findings

Homothetic parallelograms are similar to each other.

They are in perspective from the diagonals intersection point.

Formulas for area and perimeter are established.

Abstract

In this paper the concept of homothetic parallelogram is introduced. This concept is a generalization of Varignon's and Wittenbauer's parallelograms of an arbitrary quadrangle, whose diagonals are not parallel. A formula for the area and perimeter of a homothetic parallelogram for the case when quadrangles are not crossed is obtained. The fact that homothetic parallelograms are similar to one another and are in perspective from diagonals intersection point is proved.