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Cramer-Castillon on a Triangle's Incircle and Excircles

Dominique Laurain, Peter Moses, Dan Reznik

TL;DR

This work analyzes the Cramer-Castillon problem for a triangle when the circle is the triangle's incircle or excircles. It proves there are exactly two CCP solutions per circle, yielding 8 new triangles and 24 new vertices, with explicit barycentric expressions involving the golden ratio $\phi$ and a rich overlap with Brocard geometry (common circumcenter $X_3$, symmedian $X_6$, Brocard circle/axis/inellipse, isodynamic points, and Lemoine axis) that all align through the de Longchamps point $X_{mw}$ (and $X_{20}$ for excircles). A projective-invariance perspective generalizes the CCP to any inconic, providing explicit vertex formulas and showing the two-solution structure persists under perspectivity. The excircle case parallels the incircle results, with the four Brocard axes converging at $X_{20}$ and the A-excircle symmedian tied to the A-exversion of the reference's Gergonne point. Collectively, the paper reveals deep connections between CCP, Brocard geometry, and projective transformations, enabling systematic derivation of all solution vertices from a single incircle configuration.

Abstract

The Cramer-Castillon problem (CCP) consists in finding one or more polygons inscribed in a circle such that their sides pass cyclically through a list of $N$ points. We study this problem where the points are the vertices of a triangle and the circle is either the incircle or excircles.

Cramer-Castillon on a Triangle's Incircle and Excircles

TL;DR

This work analyzes the Cramer-Castillon problem for a triangle when the circle is the triangle's incircle or excircles. It proves there are exactly two CCP solutions per circle, yielding 8 new triangles and 24 new vertices, with explicit barycentric expressions involving the golden ratio and a rich overlap with Brocard geometry (common circumcenter , symmedian , Brocard circle/axis/inellipse, isodynamic points, and Lemoine axis) that all align through the de Longchamps point (and for excircles). A projective-invariance perspective generalizes the CCP to any inconic, providing explicit vertex formulas and showing the two-solution structure persists under perspectivity. The excircle case parallels the incircle results, with the four Brocard axes converging at and the A-excircle symmedian tied to the A-exversion of the reference's Gergonne point. Collectively, the paper reveals deep connections between CCP, Brocard geometry, and projective transformations, enabling systematic derivation of all solution vertices from a single incircle configuration.

Abstract

The Cramer-Castillon problem (CCP) consists in finding one or more polygons inscribed in a circle such that their sides pass cyclically through a list of points. We study this problem where the points are the vertices of a triangle and the circle is either the incircle or excircles.

Paper Structure

This paper contains 5 sections, 17 theorems, 16 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Cramer-Castillon on the Incircle
  3. Shared Brocard objects
  4. Cramer-Castillon on the Excircles
  5. Center Correspondences

Key Result

Proposition 1

The CCP on a triangle $T$ and its incircle $\mathcal{C}$ admits exactly two solutions $T_1$ and $T_2$, whose barycentric vertex matrices with respect to $T$ are given by: where $u = (s-a)$, $v = (s-b)$, $w = (s- c)$.

Figures (6)

  • Figure 1: The Cramer-Castillon problem (CCP) in the $N=3$ case. In the left (resp. right) picture, two points are exterior and one is interior (resp. all exterior) to the target circle. In each case, two solutions are shown (magenta and orange).
  • Figure 2: The two solutions of CCP (orange, magenta) on the incircle $\mathcal{C}$ of a triangle $T=ABC$. Since both are inscribed in $\mathcal{C}$, they share their circumcenter $X_3$ (at the incenter $X_{1,\text{ref}}$ of the reference). Also shared is the symmedian point $X_6$, which coincides with the Gergonne $X_{7,\text{ref}}$ of the reference.
  • Figure 3: top: The construction for the two points $H_1$ and $H_2$ which define the homography axis used in \ref{['prop:two-sols']}. The sequence $C_1, ... C_4$ is not shown, but is as $A_1 ... A_4$ and $B_1 ... B_4$. bottom: a slight zoom-in on the region where $H_1$ and $H_2$ are.
  • Figure 4: The two solutions (orange, magenta) of CCP on a triangle's incircle (black) share all Brocard geometry objects, to be sure: their Brocard points $\Omega_1,\Omega_2$, Brocard circle and axis (dashed purple), Brocard inellipse (light blue) whose foci are the Brocard points, the two isodynamic points $X_{15}$, and $X_{16}$ (not shown) and the Lemoine axis (solid purple).
  • Figure 5: Since the CCP is projectively-invariant, its solution on a $ABC$ (top) with respect to a generic inconic(black) can be regarded as the pre-image of a perspectivity $\Pi$ which sends said inconic to a circle (black, bottom). Clearly, this circle is an incircle of the new triangle $A'B'C'$. This implies that the two solutions in the original case will envelop a conic (light blue, top) which is the pre-image of the Brocard inellipse (light blue, bottom) under $\Pi$.
  • ...and 1 more figures

Theorems & Definitions (22)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Proposition 3
  • Corollary 4
  • Corollary 5
  • ...and 12 more