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:
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.