Relationships between a Central Quadrilateral and its Reference Quadrilateral

TL;DR

This paper investigates the geometric relationships between a convex quadrilateral and a derived central quadrilateral formed by triangle centers, using computational methods to analyze congruence, similarity, and other properties.

Contribution

It introduces a computational approach to analyze how various triangle centers influence the properties of the central quadrilateral relative to the reference quadrilateral.

Findings

Identifies conditions for congruence and similarity between quadrilaterals.

Analyzes relationships for special points like Steiner and Poncelet points.

Provides a comprehensive computational comparison across many configurations.

Abstract

Let P be a point inside a convex quadrilateral ABCD. The lines from P to the vertices of the quadrilateral divide the quadrilateral into four triangles. If we locate a triangle center in each of these triangles, the four triangle centers form another quadrilateral called a central quadrilateral. For each of various shaped quadrilaterals, and each of 1000 different triangle centers, we compare the reference quadrilateral to the central quadrilateral. Using a computer, we determine how the two quadrilaterals are related. For example, we test to see if the two quadrilaterals are congruent, similar, have the same area, or have the same perimeter. We also look for such relationships when P is a special point associated with the reference quadrilateral, such as being the diagonal point, Steiner point, or Poncelet point.