Heron's Formula, Descartes Circles, and Pythagorean Triangles

TL;DR

This paper explores the connections between Heron's formula, Descartes circle theorem, and Pythagorean triangles, revealing how primitive Pythagorean triples generate integral circle packings and relate to classical geometric formulas.

Contribution

It demonstrates that each primitive Pythagorean triple produces a Descartes quadruple with integral curvatures, linking Pythagorean triples to integral Apollonian packings.

Findings

Primitive Pythagorean triples generate integral Descartes quadruples.

These quadruples produce integral Apollonian packings with rectangular arrangements.

The work synthesizes new and classical results in geometry and number theory.

Abstract

This article highlights interactions of diverse areas: the Heron formula for the area of a triangle, the Descartes circle equation, and right triangles with integer or rational sides. New and old results are synthesized. We show that every primitive Pythagorean triple (PPT), furnishes a Descartes quadruple of tangent circles with integral curvatures, and so generates an integral Apollonian packing (IAP) containing a rectangle of centers. Thus Pythagorean triples serve to generate a large number of in-equivalent integral packings.