Tangential trapezoid

In Euclidean geometry, a tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the incircle or inscribed circle. It is the special case of a tangential quadrilateral in which at least one pair of opposite sides are parallel. As for other trapezoids, the parallel sides are called the bases and the other two sides the legs. The legs can be equal (see isosceles tangential trapezoid below), but they don't have to be.

Examples of tangential trapezoids are rhombi and squares.

A convex quadrilateral is tangential if and only if opposite sides satisfy Pitot's theorem:

In turn, a tangential quadrilateral is a trapezoid if and only if either of the following two properties hold (in which case they both do):

The formula for the area of a trapezoid can be simplified using Pitot's theorem to get a formula for the area of a tangential trapezoid. If the bases have lengths a and b, and any one of the other two sides has length c, then the area K is given by the formula

The area can be expressed in terms of the tangent lengths e, f, g, h as

Using the same notations as for the area, the radius in the incircle is

The diameter of the incircle is equal to the height of the tangential trapezoid.

The inradius can also be expressed in terms of the tangent lengths as

Moreover, if the tangent lengths e, f, g, h emanate respectively from vertices A, B, C, D and AB is parallel to DC, then

...
Wikipedia