Tangential Quadrilateral - Characterizations

Characterizations

In a tangential quadrilateral, the four angle bisectors meet at the center of the incircle. Conversely, a convex quadrilateral in which the four angle bisectors meet at a point must be tangential and the common point is the incenter.

According to the Pitot theorem, the two pairs of opposite sides in a tangential quadrilateral add up to the same total length, which equals the semiperimeter s of the quadrilateral:

Conversely a convex quadrilateral in which a + c = b + d must be tangential.

If opposite sides in a convex quadrilateral ABCD (that is not a trapezoid) intersect at E and F, then it is tangential if and only if either of

or

The second of these is almost the same as one of the equalities in Urquhart's theorem. The only differences are the signs on both sides; in Urquhart's theorem there are sums instead of differences.

Another necessary and sufficient condition is that a convex quadrilateral ABCD is tangential if and only if the incircles in the two triangles ABC and ADC are tangent to each other.

A characterization regarding the angles formed by diagonal BD and the four sides of a quadrilateral ABCD is due to Iosifescu. He proved in 1954 that a convex quadrilateral has an incircle if and only if

Further, a convex quadrilateral with successive sides a, b, c, d is tangential if and only if

where R_a, R_b, R_c, R_d are the radii in the circles externally tangent to the sides a, b, c, d respectively and the extensions of the adjacent two sides for each side.

Several more characterizations are known in the four subtriangles formed by the diagonals.