Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

For the same reason the parallelogram KM is also equal and similar to CR, and further EP to DF; therefore three parallelograms of the solid KP are equal and similar to three parallelograms of the solid CD.

But the former three parallelograms are equal and similar to their opposites, and the latter three to their opposites; [XI. 24] therefore the whole solid KP is equal and similar to the whole solid CD. [XI. Def. 10]

Let the parallelogram GK be completed, and on the parallelograms GK, KL as bases, and with the same height as that of AB, let the solids EO, LQ be completed.

Then since; owing to the similarity of the solids AB, CD, as AE is to CF, so is EG to FN, and EH to FR, while CF is equal to EK, FN to EL, and FR to EM, therefore, as AE is to EK, so is GE to EL, and HE to EM.

But, as AE is to EK, so is AG to the parallelogram GK, as GE is to EL, so is GK to KL, and, as HE is to EM, so is QE to KM; [VI. 1] therefore also, as the parallelogram AG is to GK, so is GK to KL, and QE to KM.

But, as AG is to GK, so is the solid AB to the solid EO, as GK is to KL, so is the solid OE to the solid QL, and, as QE is to KM, so is the solid QL to the solid KP; [XI. 32] therefore also, as the solid AB is to EO, so is EO to QL, and QL to KP.

But, if four magnitudes be continuously proportional, the first has to the fourth the ratio triplicate of that which it has to the second; [V. Def. 10] therefore the solid AB has to KP the ratio triplicate of that which AB has to EO.

But, as AB is to EO, so is the parallelogram AG to GK, and the straight line AE to EK [VI. 1];