Dynamics of ellipses inscribed in triangles

TL;DR

This paper investigates the existence and number of ellipses inscribed in a triangle passing through given points, revealing conditions and counts for various configurations including points on the boundary and with specified slopes.

Contribution

It provides a comprehensive analysis of inscribed ellipses passing through interior and boundary points, including the impact of slope constraints, with explicit counts and conditions.

Findings

Four ellipses pass through two interior points outside certain line segments.

The number of such ellipses varies when points are on the boundary.

Some slope conditions prevent the existence of an inscribed ellipse passing through a point.

Abstract

Suppose that we are given two distinct points, $P_1$ and $P_2$, in the interior of a triangle, $T$. Is there always an ellipse inscribed in $T$ which also passes through $P_1$ and $P_2$ ? If yes, how many such ellipses ? We answer those questions in this paper. It turns out that, except for $P_1$ and $P_2$ on a union of three line segments, there are four such ellipses which pass through $P_1$ and $P_2$. We also answer a similar question if instead $P_1$ and $P_2$ lie on the boundary of $T$. Finally, an interesting related question, is the following: Given a point, $P$, in the interior of a triangle, $T$, and a real number, $r$, is there always an ellipse inscribed in $T$ which passes through $P$ and has slope $r$ at $P$ ? Again, if yes, how many such ellipses ? The answer is somewhat different than for the two point case without specifying a slope. There are cases where no such ellipse exists.