Ex-tangential quadrilateral

In <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a>, an ex-tangential quadrilateral is a <a href="/facts/Convex_polygon/7x35w610">convex</a> <a href="/facts/Quadrilateral/QCGqrkp2">quadrilateral</a> where the extensions of all four sides are tangent to a <a href="/facts/Circle/9fy5ggKn">circle</a> outside the quadrilateral. It has also been called an exscriptible quadrilateral. The circle is called its excircle, its radius the exradius and its center the excenter (E in the figure). The excenter lies at the intersection of six angle bisectors. These are the internal <a href="/facts/Angle_bisector/JTsya7Ca">angle bisectors</a> at two opposite vertex angles, the <a href="/facts/Internal_and_external_angle/jd78Qkbt">external angle</a> bisectors (<a href="/facts/Supplementary_angle/gFUTEQYq">supplementary angle</a> bisectors) at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect (see the figure to the right, where four of these six are dotted line segments). The ex-tangential quadrilateral is closely related to the <a href="/facts/Tangential_quadrilateral/JBdfotIx">tangential quadrilateral</a> (where the four sides are tangent to a circle).
Another name for an excircle is an escribed circle, but that name has also been used for a circle tangent to one side of a convex quadrilateral and the extensions of the adjacent two sides. In that context all convex quadrilaterals have four escribed circles, but they can at most have one excircle.

Ex-tangential quadrilateral open-in-new

Ex-tangential quadrilateral