# When is an ellipse inscribed in a quadrilateral tangent at the midpoint of two or more sides ?

## TL;DR

This paper investigates conditions under which an inscribed ellipse in a quadrilateral is tangent at midpoints of sides, revealing that such ellipses are only possible in specific quadrilaterals like trapezoids or midpoint diagonal quadrilaterals.

## Contribution

It strengthens previous results by proving that non-parallelogram quadrilaterals cannot have an inscribed ellipse tangent at three midpoints, and characterizes quadrilaterals with ellipses tangent at two midpoints.

## Key findings

- No inscribed ellipse tangent at three midpoints in non-parallelogram quadrilaterals.
- Quadrilaterals with ellipses tangent at two midpoints are trapezoids or midpoint diagonal quadrilaterals.
- Ellipses tangent at midpoints are highly restricted by quadrilateral geometry.

## Abstract

In "Quartic Coincidences and the Singular Value Decomposition" by Clifford and Lachance, Mathematics Magazine, December, 2013, it was shown that if there is a midpoint ellipse(an ellipse inscribed in a quadrilateral, $Q$, which is tangent at the midpoints of all four sides of $Q$), then $Q$ must be a parallelogram. We strengthen this result by showing that if $Q$ is not a parallelogram, then there is no ellipse inscribed in $Q$ which is tangent at the midpoint of three sides of $Q$. Second, the only quadrilaterals which have inscribed ellipses tangent at the midpoint of even two sides of $Q$ are trapezoids or what we call a midpoint diagonal quadrilateral(the intersection point of the diagonals of $Q$ coincides with the midpoint of at least one of the diagonals of $Q$).