Dynamics of ellipses inscribed in quadrilaterals

TL;DR

This paper investigates the number and properties of ellipses inscribed in convex quadrilaterals passing through specific interior points, revealing precise conditions based on the point’s location relative to the diagonals and boundary.

Contribution

It provides a complete characterization of the existence and uniqueness of inscribed ellipses passing through interior points of convex quadrilaterals, depending on their position.

Findings

Two inscribed ellipses pass through points not on diagonals.

One inscribed ellipse passes through points on diagonals (excluding intersection).

No inscribed ellipse passes through the intersection point of diagonals.

Abstract

Let Q be a convex quadrilateral in the xy plane and let int(Q) denote the interior of Q. Let D_1 and D_2 denote the diagonals of Q and let P denote their point of intersection. For (i)-(iii), let P_0 = (x_0,y_0) be a point in the interior of Q. We prove the following: (i) If P_0 does not lie on D_1 or on D_2, then there are exactly two ellipses inscribed in Q which pass through P_0. (ii) If P_0 does lie on D_1 or on D_2, but does not equal P, then there is exactly one ellipse inscribed in Q which passes through P_0. (iii) There is no ellipse inscribed in Q which passes through P. (iv) If P_0 lies on the boundary of Q, but P_0 is not one of the vertices of Q, then there is exactly one ellipse inscribed in Q which passes through P_0(and is thus tangent to Q at one of its sides).