Writing equations for similar triangles

It is often difficult to visually recognise the corresponding parts of similar figures, especially in the case of overlapping triangles. It is therefore necessary to develop a formal approach to the analysis of similar figures, i.e. a method of comparing corresponding parts without visual reference to the figures.

It can be useful to use colours and signs to interpret and make sense of sketches, and/or redraw the triangles, separating them into two separate figures.

A good way to write down the order and subsequent proportional equations is to use the “umbrella method” of naming similar triangles.
Note: The umbrella method also applies to congruent triangles, where corresponding lengths are equal!

In the sketch below we know that ÐA = ÐD, ÐB = ÐF and ÐC = ÐE. In the umbrella method we write the two triangle names strictly in the order of equality:
DABC ||| DDFE.

From this order we can then deduce the appropriate proportional equations in an automatic way without visual reference to the figures, as illustrated below: