An Infinite, Converging, Sequence of Brocard Porisms

TL;DR

This paper introduces an infinite, converging sequence of Brocard porisms generated through a recursive process, revealing new geometric properties and their embedding in a continuous family of porisms.

Contribution

It presents a novel recursive construction of Brocard porisms leading to an infinite converging sequence and explores their relationship within a continuous family.

Findings

An infinite sequence of Brocard porisms converges.

The sequence is embedded in a continuous family of porisms.

The Brocard angle remains invariant throughout the sequence.

Abstract

The Brocard porism is a known 1d family of triangles inscribed in a circle and circumscribed about an ellipse. Remarkably, the Brocard angle is invariant and the Brocard points are stationary at the foci of the ellipse. In this paper we show that a certain derived triangle spawns off a second, smaller, Brocard porism so that repeating this calculation produces an infinite, converging sequence of porisms. We also show that this sequence is embedded in a continuous family of porisms.