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A generalization of parallelograms invol...
Horwitz, Alan...

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A generalization of parallelograms involving inscribed ellipses, conjugate diameters, and tangency chords by Horwitz, Alan ( Author )
Lemonjar Open Access
Australian National University
09-08-2023
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 pair of conjugate diameters parallel to the diagonals of $Q$. Finally, we show that there is a unique ellipse, $E_I$, of minimal eccentricity incribed in a midpoint diagonal quadrilateral, $Q$, and we show that the equal conjugate diameters of $E_I$ are parallel to the diagonals of $Q$.
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Article
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30.00 KB
English
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MYR 0.01
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http://arxiv.org/abs/1610.06037
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