An example:
Above we have three similar right triangles, so we write down the sides in order from smallest to largest, and then repeat, going through each triangle in order.
The trick is that any four entries making a rectangle are a valid ratio from this data.
In particular, I'm hoping you may be able to see a quick proof of Pythagoras's Theorem.
There are several more examples. The most complicated is one from Inversive Transformation in a circle. The rule for the transformation is OA times OA' = r^2, where r is the radius of the circle with the solid line.
As we work through the example, you should be able to see how the ratio boxes dramatically simplify the bookkeeping involved in the proof. The chapter is on my Dropbox as a pdf.
The theorem is one of my very favorites.
