Lattice Equable Quadrilaterals III: tangential and extangential cases

TL;DR

This paper classifies all lattice equable quadrilaterals with tangential and extangential properties, revealing a finite set of convex cases and infinite families of concave ones through Diophantine equations.

Contribution

It provides a complete classification of lattice equable quadrilaterals in tangential and extangential cases, identifying finite and infinite families using Diophantine equations.

Findings

6 convex tangential lattice equable quadrilaterals up to Euclidean motions

7 infinite families of concave tangential lattice equable quadrilaterals

Only one concave extangential lattice equable quadrilateral besides kites

Abstract

A lattice equable quadrilateral is a quadrilateral in the plane whose vertices lie on the integer lattice and which is equable in the sense that its area equals its perimeter. This paper treats the tangential and extangential cases. We show that up to Euclidean motions, there are only 6 convex tangential lattice equable quadrilaterals, while the concave ones are arranged in 7 infinite families, each being given by a well known diophantine equation of order 2 in 3 variables. On the other hand, apart from the kites, up to Euclidean motions there is only one concave extangential lattice equable quadrilateral, while there are infinitely many convex ones.