An ellipse inscribed in quadrilateral Q has one focus equidistant from the vertices if and only if Q is orthodiagonal, with the other focus at the diagonal intersection.
Bond, ”A New Algorithm for Scan Conversion of a General Ellipse”, preprint, http://www.crbond.com/papers/ell alg.pdf
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Besant quadrilaterals
An ellipse inscribed in quadrilateral Q has one focus equidistant from the vertices if and only if Q is orthodiagonal, with the other focus at the diagonal intersection.