Ex-tangential quadrilateral

Parallelograms (which include squares, rhombi, and rectangles) can be considered ex-tangential quadrilaterals with infinite exradius since they satisfy the characterizations in the next section, but the excircle cannot be tangent to both pairs of extensions of opposite sides (since they are parallel).

If opposite sides in a convex quadrilateral ABCD intersect at E and F, then The implication to the right is named after L. M. Urquhart (1902–1966) although it was proved long before by Augustus De Morgan in 1841.

Daniel Pedoe named it the most elementary theorem in Euclidean geometry since it only concerns straight lines and distances.

[6] That there in fact is an equivalence was proved by Mowaffac Hajja,[6] which makes the equality to the right another necessary and sufficient condition for a quadrilateral to be ex-tangential.

[4] Thus a convex quadrilateral has an incircle or an excircle outside the appropriate vertex (depending on the column) if and only if any one of the five necessary and sufficient conditions below is satisfied.

But when solving for x, we must choose the other root of the quadratic equation for the ex-bicentric quadrilateral compared to the bicentric.