The Shape of Central Quadrilaterals

TL;DR

This paper investigates the geometric properties of quadrilaterals by examining triangle centers within their component triangles, revealing conditions under which these centers form special quadrilaterals like rhombi or cyclic quadrilaterals.

Contribution

It introduces a computational approach to identify when triangle centers in quadrilaterals form specific types of quadrilaterals, expanding understanding of geometric configurations.

Findings

Diagonals of equidiagonal quadrilaterals divide them into four triangles.

Nagel points of these triangles form an orthodiagonal quadrilateral.

Computational methods reveal conditions for special quadrilateral formations.

Abstract

The diagonals of a quadrilateral form four component triangles (in two ways). For each of various shaped quadrilaterals, we examine 1000 triangle centers located in these four component triangles. Using a computer, we determine when the four centers form a special quadrilateral, such as a rhombus or a cyclic quadrilateral. A typical result is the following. The diagonals of an equidiagonal quadrilateral divide the quadrilateral into four nonoverlapping triangles. Then the Nagel points of these four triangles form an orthodiagonal quadrilateral.