Let there be pyramids of the same height with the polygonal bases ABCDE and FGHKL and vertices M and N.
I say that the base ABCDE is to the base FGHKL as the pyramid ABCDEM is to the pyramid FGHKLN.
Join AC, AD, FH, and FK.
Since then ABCM and ACDM are two pyramids with triangular bases and equal height, therefore they are to one another as their bases. Therefore the base ABC is to the base ACD as the pyramid ABCM is to the pyramid ACDM. And, taken together, the base ABCD is to the base ACD as the pyramid ABCDM is to the pyramid ACDM.
But the base ACD is to the base ADE as the pyramid ACDM is to the pyramid ADEM.
Therefore, ex aequali, the base ABCD is to the base ADE as the pyramid ABCDM is to the pyramid ADEM.
And again, taken together, the base ABCDE is to the base ADE as the pyramid ABCDEM is to the pyramid ADEM. Similarly also it can be proved that the base FGHKL is to the base FGH as the pyramid FGHKLN is to the pyramid FGHN.
And, since ADEM and FGHN are two pyramids with triangular bases and equal heights, therefore the base ADE is to the base FGH as the pyramid ADEM is to the pyramid FGHN.
But the base ADE is to the base ABCDE as the pyramid ADEM is to the pyramid ABCDEM. Therefore, ex aequali, the base ABCDE is to the base FGH as the pyramid ABCDEM is to the pyramid FGHN.
But further the base FGH is to the base FGHKL as the pyramid FGHN is to the pyramid FGHKLN. Therefore also, ex aequali, the base ABCDE is to the base FGHKL as the pyramid ABCDEM is to the pyramid FGHKLN.
Therefore, pyramids of the same height with polygonal bases are to one another as their bases.