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]
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]
Again, since, as DC is to CE, so is the parallelogram CH to CF, [VI. 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]
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.
Complete text
Book I