An Infinite, Converging, Sequence of Brocard Porisms
This is an incremental result in geometry, addressing a specific problem for researchers in triangle geometry and porisms.
The paper tackles the problem of extending the Brocard porism by demonstrating that a derived triangle generates an infinite, converging sequence of porisms, and shows this sequence is part of a continuous family.
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.