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arXiv · 2102.11369

Midpoint Diagonal Quadrilaterals

Alan Horwitz
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Abstract

A convex quadrilateral, $Q$, is called a midpoint diagonal quadrilateral if the intersection point of the diagonals of $Q$ coincides with the midpoint of at least one of the diagonals of $Q$. A parallelogram, P, is a special case of a midpoint diagonal quadrilateral since the diagonals of P bisect one another. We prove two results about ellipses inscribed in midpoint diagonal quadrilaterals, which generalize properties of ellipses inscribed in parallelograms involving convex quadrilaterals. First, $Q$ is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in $Q$ has tangency chords which are parallel to one of the diagonals of $Q$. Second, $Q$ is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in $Q$ has a unique pair of conjugate diameters parallel to the diagonals of $Q$. Finally, we show that there is a unique ellipse, $E_I$, of minimal eccentricity inscribed in a midpoint diagonal quadrilateral, $Q$, and also that the unique pair of conjugate diameters parallel to the diagonals of $Q$ are the equal conjugate diameters of $E_I$.

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Alan Horwitz. 2021-02-22. Midpoint Diagonal Quadrilaterals. https://arxiv.org/abs/2102.11369

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