More Relationships between a Central Quadrilateral and its Reference Quadrilateral

TL;DR

This paper explores the geometric relationships between a reference quadrilateral and a derived central quadrilateral formed by specific triangle centers within its half triangles, using computational methods to analyze various properties.

Contribution

It introduces a systematic computational approach to compare reference and central quadrilaterals across different shapes and triangle centers, revealing new geometric relationships.

Findings

Identifies conditions for congruence and similarity between quadrilaterals.

Provides a comprehensive computational analysis of 1000 triangle centers.

Discovers patterns in area and perimeter relationships.

Abstract

The diagonals of a quadrilateral form four associated triangles, called half triangles. Each half triangle is bounded by two sides of the quadrilateral and one diagonal. If we locate a triangle center (such as the incenter, centroid, orthocenter, etc.) 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.