Besant quadrilaterals

We solve the following problem of W.H. Besant using a formula for the coefficients of an ellipse inscribed in a quadrilateral, $Q$: \enquote{If an ellipse be inscribed in a quadrilateral so that one focus is equidistant from the four vertices(call that point $EP$), the other focus must be at the intersection of the diagonals(call that point $IP$).} We also prove somewhat more than just solving Besant's problem itself, though it would be nice to see the details of the geometric approach proposed by Besant. More precisely, we also prove the converse result and additional results when $Q$ is a trapezoid. Finally, we show that such an inscribed ellipse exists if and only if $Q$ is orthodiagonal.