For let them be placed so that BC is in a straight line with CG ; therefore DC is also in a straight line with CE .

Let the parallelogram DG be completed; let a straight line K be set out, and let it be contrived that, as BC is to CG , so is K to L , and, as DC is to CE , so is L to M . [[VI. 12](elem.6.12)]

Then the ratios of K to L and of L to M are the same as the ratios of the sides, namely of BC to CG and of DC to CE .

But the ratio of K to M is compounded of the ratio of K to L and of that of L to M ; so that K has also to M the ratio compounded of the ratios of the sides.

Now since, as BC is to CG , so is the parallelogram AC to the parallelogram CH , [[VI. 1](elem.6.1)] while, as BC is to CG , so is K to L , therefore also, as K is to L , so is AC to CH . [[V. 11](elem.5.11)]

Again, since, as DC is to CE , so is the parallelogram CH to CF , [[VI. 1](elem.6.1)] while, as DC is to CE , so is L to M , therefore also, as L is to M , so is the parallelogram CH to the parallelogram CF . [[V. 11](elem.5.11)]

Since then it was proved that, as K is to L , so is the parallelogram AC to the parallelogram CH , and, as L is to M , so is the parallelogram CH to the parallelogram CF , therefore, ex aequali , as K is to M , so is AC to the parallelogram
CF .

But K has to M the ratio compounded of the ratios of the sides; therefore AC also has to CF the ratio compounded of the ratios of the sides.

Therefore etc. Q. E. D.

the ratio compounded of the ratios of the sides, λόγον τὸν συγκείμενον ἐκ τῶν πλευρῶν which, meaning literally the ratio compounded of the sides ,

is negligently written here and commonly for λόγον τὸν συγκείμενον ἐκ τῶν τῶν πλευρῶν (sc. λόγων ).

let it be contrived that, as BC is to CG, so is K to L. The Greek phrase is of the usual terse kind, untranslatable literally : καὶ γεγονέτω ὡς μὲν ἡ ΒΓ πρὸς τὴν ΓΗ, οὕτως ἡ Κ πρὸς τὸ Λ , the words meaning and let (there) be made, as BC to CG , so K to L ,

where L is the straight line which has to be constructed.