For since, as BG is to C, so is EH to F, inversely, as C is to BG, so is F to EH.

Since, then, as AB is to C, so is DE to F, and, as C is to BG, so is F to EH, therefore, ex aequali, as AB is to BG, so is DE to EH. [[V. 22](elem.5.22)]

And, since the magnitudes are proportional separando, they will also be proportional componendo; [[V. 18](elem.5.18)] therefore, as AG is to GB, so is DH to HE.

But also, as BG is to C, so is EH to F; therefore, ex aequali, as AG is to C, so is DH to F. [[V. 22](elem.5.22)]

Therefore etc. Q. E. D.