An App for the Discovery of Properties of Poncelet Triangles

TL;DR

The paper presents a free, web-based app that enables interactive, real-time exploration of Poncelet triangle families bounded by two ellipses, with emphasis on the loci of triangle centers and derived points. It combines extensive geometric functionality—covering standard and exotic triangles, numerous cevian/pedal constructions, inversive and polar transforms, envelopes, and central/bicentric configurations—with automatic conic-detection and invariants reporting to reveal underlying structure. A key contribution is the large, sharable experiment library (including hundreds of playlists and thousands of parameter combinations) and a UI designed for rapid, visual hypothesis testing, enabling researchers to generate, test, and share conjectures via URLs and export formats. The tool thus advances experimental geometry in the dynamic setting of Poncelet porisms, enabling systematic observation of conic vs non-conic loci and invariant quantities, and providing an accessible platform for education, exploration, and collaboration.

Abstract

We describe a newly-developed, free, browser-based application, for the interactive exploration of the dynamic geometry of Poncelet families of triangles. The main focus is on responsive display of the beauteous loci of centers of such families, refreshing them smoothly upon any changes in simulation parameters. The app informs the user when curves swept are conics and reports if certain metric quantities are conserved. Live simulations can be easily shared via a URL. A list of more than 400 pre-made experiments is included which can be regarded as conjectures and/or exercises. Millions of experiment combinations are possible.

Figures (30)

• Figure 1: Locus Visualization app to explore 3-periodic families. Shown are the loci of $X_k$, $k=$1,2,3,4, over billiard 3-periodics. The "(E)" suffix indicated they are numerically ellipses. https://bit.ly/3yV8caF; Also see our tutorial https://bit.ly/3iCKHxn.
• Figure 2: Four examples of the kinds of color-filled loci which can be produced with the app. https://bit.ly/34sO8Px and https://youtu.be/l-O5UT8tpuw
• Figure 3: The confocal family is shown at the top left. Also shown are 5 other "famous" concentric families. https://youtu.be/14TQ5WlZxUw
• Figure 4: Four circle-inscribed families supported: inellipse (top-left): "inellipse", caustic is a concentric inellipse (appears in \ref{['fig:six-caps']}); bicentric (top-right): Chapple's porism, caustic is the fixed incircle; macbeath (bottom-left), excentrals to the bicentric family, caustic is the MacBeath inconic mw, with foci on $X_3$ and $X_4$; brocard (bottom-right): also known as the Brocard porism bradley2011-brocard, or the "harmonic family", caustic is the Brocard inellipse mw, with foci on the two Brocard points $\Omega_1,\Omega_2$.
• Figure 5: Four ellipse-inscribed families with non-concentric, circular caustics: top-left: stationary incenter $X_1$ (at a focus); top-right: stationary barycenter $X_2$ (and Nagel point $X_8$ at center); bottom-left: stationary orthocenter $X_4$ (at a focus); bottom-right: [to do] stationary Gergonne point $X_7$. Caustic centers and radii kindly derived by Ronaldo Garcia garcia2024-private.