Lattice equable quadrilaterals II -- kites, trapezoids and cyclic quadrilaterals

TL;DR

This paper classifies lattice equable quadrilaterals, including kites, trapezoids, and cyclic types, revealing their infinite families, finite classifications, and properties of their diagonals, with connections to Pell equations.

Contribution

It provides a complete classification of lattice equable kites, trapezoids, and cyclic quadrilaterals, identifying infinite families and finite sets with specific properties.

Findings

4 infinite families of lattice equable kites from Pell equations

Exactly 5 lattice equable trapezoids (including special types)

4 lattice equable cyclic quadrilaterals

Abstract

We show that there are 4 infinite families of lattice equable kites, given by corresponding Pell or Pell-like equations, but up to Euclidean motions, there are exactly 5 lattice equable trapezoids (2 isosceles, 2 right, 1 singular) and 4 lattice equable cyclic quadrilaterals. We also show that, with one exception, the interior diagonals of lattice equable quadrilaterals are irrational.