Tangential Quadrilateral - Inradius

Inradius

The inradius in a tangential quadrilateral with consecutive sides a, b, c, d is given by

where K is the area of the quadrilateral and s is its semiperimeter. For a tangential quadrilateral with given sides, the inradius is maximum when the quadrilateral is also cyclic (and hence a bicentric quadrilateral).

In terms of the tangent lengths, the incircle has radius

The inradius can also be expressed in terms of the distances from the incenter I to the vertices of the tangential quadrilateral ABCD. If u = AI, v = BI, x = CI and y = DI, then

where .