Schellbach-style Formulae for the Derousseau-Pampuch Generalizations of the Malfatti Circles

TL;DR

This paper derives explicit formulas for the radii of special tangent circle triplets within a triangle, extending Malfatti circle generalizations using trigonometric and hyperbolic functions.

Contribution

It provides explicit Schellbach-style formulas for the 32 triplets of tangent circles in a triangle, generalizing the De Rousseau-Pampuch configurations.

Findings

Formulas for circle radii derived from triangle side lengths.

Explicit expressions using trigonometric and hyperbolic functions.

Applicable to all 32 tangent circle triplet configurations.

Abstract

It is known that there exist 32 triplets of circles such that each circle is tangent to the other two circles and to two of the sides of the triangle or their extensions. We provide formulae to obtain the radii of the circles for each of the 32 triplets from the side lengths of the reference triangle by means of trigonometric or hyperbolic functions.