PROPOSITION 22.

If four straight lines be proportional, the rectilineal figures similar and similarly described upon them will also be proportional; and, if the rectilineal figures similar and similarly described upon them be proportional, the straight lines will themselves also be proportional.

Let the four straight lines AB, CD, EF, GH be proportional, so that, as AB is to CD, so is EF to GH, and let there be described on AB, CD the similar and similarly situated rectilineal figures KAB, LCD, and on EF, GH the similar and similarly situated rectilineal figures MF, NH; I say that, as KAB is to LCD, so is MF to NH.

For let there be taken a third proportional O to AB, CD, and a third proportional P to EF, GH. [VI. 11]

Then since, as AB is to CD, so is EF to GH,

and, as CD is to O, so is GH to P,
therefore, ex aequali, as AB is to O, so is EF to P. [V. 22]

But, as AB is to O, so is KAB to LCD, [VI. 19, Por.]

and, as EF is to P, so is MF to NH;
therefore also, as KAB is to LCD, so is MF to NH. [V. 11]

Next, let MF be to NH as KAB is to LCD; I say also that, as AB is to CD, so is EF to GH.

For, if EF is not to GH as AB to CD,

let EF be to QR as AB to CD, [VI. 12]
and on QR let the rectilineal figure SR be described similar and similarly situated to either of the two MF, NH. [VI. 18]

Since then, as AB is to CD, so is EF to QR, and there have been described on AB, CD the similar and similarly situated figures KAB, LCD, and on EF, QR the similar and similarly situated figures MF, SR, therefore, as KAB is to LCD, so is MF to SR.

But also, by hypothesis,

as KAB is to LCD, so is MF to NH; therefore also, as MF is to SR, so is MF to NH. [V. 11]

Therefore MF has the same ratio to each of the figures NH, SR;

therefore NH is equal to SR. [V. 9]

But it is also similar and similarly situated to it;

therefore GH is equal to QR.

And, since, as AB is to CD, so is EF to QR, while QR is equal to GH, therefore, as AB is to CD, so is EF to GH.

Therefore etc. Q. E. D.