Variations on a Japanese Temple theorem

Extract

The following Japanese Temple geometry theorem appears in Fukagawa and Rigby [1]:

Thoerem 1: The products of the radii of the excircles on each pair of opposite sides of a circumscribed quadrilateral are equal.

See Figure 1. A circumscribed quadrilateral (also called a tangential quadrilateral) is one that possesses an inscribed circle. Now the diagram contains four lines (the sides of the quadrilateral) and one circle S (the inscribed circle) that touches all four of them; this we call the base circle of the theorem. The other four circles each touch just three of the lines. For the moment, assume no two of these lines are parallel, and also assume that no circle apart from S touches all four lines; the exceptional cases will be discussed in Section 4. So any three of the lines are sides of a triangle.