# Brahmagupta quadrilaterals with equal perimeters and equal areas

## TL;DR

This paper investigates the problem of finding pairs of Brahmagupta quadrilaterals with equal perimeters and areas, providing parametric solutions and expanding the known set of such quadrilaterals.

## Contribution

It introduces the first parametric solutions for Brahmagupta quadrilaterals with equal perimeters and areas, including cases with both equal and unequal sides.

## Key findings

- Two parametric solutions for Brahmagupta quadrilaterals with equal perimeter and area.
- First solution produces quadrilaterals with two equal sides.
- Second solution produces quadrilaterals with all sides unequal.

## Abstract

A cyclic quadrilateral is called a Brahmagupta quadrilateral if its four sides, the two diagonals and the area are all given by integers. In this paper we consider the hitherto unsolved problem of finding two Brahmagupta quadrilaterals with equal perimeters and equal areas. We obtain two parametric solutions of the problem -- the first solution generates examples in which each quadrilateral has two equal sides while the second solution gives quadrilaterals all of whose sides are unequal. We also show how more parametric solutions of the problem may be obtained.