If four straight lines are proportional, then the rectilinear figures similar and similarly described upon them are also proportional; and, if the rectilinear figures similar and similarly described upon them are proportional, then the straight lines are themselves also proportional.

Let the four straight lines *AB, CD, EF,* and *GH* be proportional, so that *AB* is to *CD* as *EF* is to *GH.* Let the similar and similarly situated rectilinear figures *KAB* and *LCD* be described on *AB* and *CD,* and the similar and similarly situated rectilinear figures *MF* and *NH* be described on *EF* and *GH.*

I say that *KAB* is to *LCD* as *MF* is to *NH.*

Take a third proportional *O* to *AB* and *CD,* and a third proportional *P* to *EF* and *GH.*

Then since *AB* is to *CD* as *EF* is to *GH,* therefore *CD* is to *O* as *GH* is to *P.* Therefore, *ex aequali, AB* is to *O* as *EF* is to *P.*

But *AB* is to *O* as *KAB* is to *LCD,* and *EF* is to *P* as *MF* is to *NH,* therefore *KAB* is to *LCD* also as *MF* is to *NH.*

Next, let *KAB* be to *LCD* as *MF* is to *NH.*

I say also that *AB* is to *CD* as *EF* is to *GH.*

For, if *EF* is not to *GH* as *AB* is to *CD,* let *EF* be to *QR* as *AB* is to *CD.* Describe the rectilinear figure *SR* similar and similarly situated to either of the two *MF* or *NH* on *QR.*

Above

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

But also, by hypothesis, *KAB* is to *LCD* as *MF* is to *NH,* therefore also *MF* is to *SR* as *MF* is to *NH.*

Therefore *MF* has the same ratio to each of the figures *NH* and *SR,* therefore *NH* equals *SR.*

But it is also similar and similarly situated to it, therefore *GH* equals *QR.*

And, since *AB* is to *CD* as *EF* is to *QR,* while *QR* equals *GH,* therefore *AB* is to *CD* as *EF* is to *GH.*

Therefore, *if four straight lines are proportional, then the rectilinear figures similar and similarly described upon them are also proportional; and, if the rectilinear figures similar and similarly described upon them are proportional, then the straight lines are themselves also proportional.*

Q.E.D.