If we label the distances AP, PB, CQ and QB as shown above then we can write three equations for the areas of the triangles as follows:

Notice that the two ratios are the same! If we call this ratio *r* then we can calculate a polynomial for *r* as follows:

Taking the positive root gives us the golden ratio:

## Subscripts

For explaining a solution like this one that uses very few variables, it's much easier to read if the variables are named with different letters, but no subscripts.