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Ex-tangential quadrilateral
In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the ''extensions'' of all four sides are tangent to a circle outside the quadrilateral.〔Radic, Mirko; Kaliman, Zoran and Kadum, Vladimir, "A condition that a tangential quadrilateral is also a chordal one", ''Mathematical Communications'', 12 (2007) pp. 33–52.〕 It has also been called an exscriptible quadrilateral.〔Bogomolny, Alexander, "Inscriptible and Exscriptible Quadrilaterals", ''Interactive Mathematics Miscellany and Puzzles'', (). Accessed 2011-08-18.〕 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 angle bisectors at two opposite vertex angles, the external angle bisectors (supplementary angle 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 tangential quadrilateral (where the four sides are tangent to a circle).
Another name for an excircle is an escribed circle,〔K. S. Kedlaya, ''Geometry Unbound'', 2006, ()〕 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.〔
==Special cases==
Kites are examples of ex-tangential quadrilaterals. 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).〔 Convex quadrilaterals whose side lengths form an arithmetic progression are always ex-tangential as they satisfy the characterization below for adjacent side lengths.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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