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.
Besant, Conic Sections Treated Geometrically, George Bell and Sons Educational Catalogue, Project Gutenberg ebook number 29913
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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.