The Diagonal Point Triangle Revisited

We derive a formula for the area of the diagonal point triangle be- longing to a tangential quadrilateral in terms of the four tangent lengths, and prove a characterization for a tangential trapezoid. The significance of the diagonal point triangle is most evident in projective ge- ometry, where it is studied in connection with the complete quadrilateral. It is for instance a well known property that the diagonal point triangle associated with a cyclic quadrilateral is self-conjugate. In (5) we derived a formula for the area of the diagonal point triangle belonging to a cyclic quadrilateral in terms of the four sides. In this note we shall derive a formula for this triangle area in connection with a tangential quadrilateral (a quadrilateral with an incircle), but here it will be in terms of the tangent lengths instead. The tangent lengths e, f , g, h in a tangential quadrilateral are defined to be the distances from the vertices to the points where the incircle is tangent to the sides (see Figure 2).