Related by Similarity II: Poncelet 3-Periodics in the Homothetic Pair and the Brocard Porism
This work addresses geometric problems in billiard dynamics and poristic triangles, offering incremental insights into similarity relationships in Poncelet families.
The paper investigates two new families of Poncelet 3-periodic triangles related by similarity transformations, specifically those inscribed in concentric homothetic ellipses and the Brocard porism with a fixed circumcircle and Brocard inellipse, finding that they share invariants like the Brocard angle and sum of cotangents.
Previously we showed the family of 3-periodics in the elliptic billiard (confocal pair) is the image under a variable similarity transform of poristic triangles (those with non-concentric, fixed incircle and circumcircle). Both families conserve the ratio of inradius to circumradius and therefore also the sum of cosines. This is consisten with the fact that a similarity preserves angles. Here we study two new Poncelet 3-periodic families also tied to each other via a variable similarity: (i) a first one interscribed in a pair of concentric, homothetic ellipses, and (ii) a second non-concentric one known as the Brocard porism: fixed circumcircle and Brocard inellipse. The Brocard points of this family are stationary at the foci of the inellipse. A key common invariant is the Brocard angle, and therefore the sum of cotangents. This raises an interesting question: given a non-concentric Poncelet family (limited or not to the outer conic being a circle), can a similar doppelgänger always be found interscribed in a concentric, axis-aligned ellipse and/or conic pair?