| Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides. | ||
| Let AC and CF be equiangular parallelograms having the angle BCD equal to the angle ECG.
I say that the parallelogram AC has to the parallelogram CF the ratio compounded of the ratios of the sides. | ||
| Let them be placed so that BC is in a straight line with CG. Then DC is also in a straight line with CE. | I.14 | |
| Complete the parallelogram DG. Set out a straight line K, and make it so that BC is to CG as K is to L, and DC is to CE as L is to M. | I.31 | |
| 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 BC is to CG as the parallelogram AC is to the parallelogram CH, and BC is to CG as K is to L, therefore K is to L as AC is to CH. | VI.1
V.11 | |
| Again, since DC is to CE as the parallelogram CH is to CF, and DC is to CE as L is to M, therefore L is to M as the parallelogram CH is to the parallelogram CF. | VI.1
V.11 | |
| Since then it was proved that K is to L as the parallelogram AC is to the parallelogram CH, and L is to M as the parallelogram CH is to the parallelogram CF, therefore, ex aequali K is to M as AC is to the parallelogram CF. | V.22 | |
| 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, equiangular parallelograms have to one another the ratio compounded of the ratios of their sides. | ||
| Q.E.D. | ||
Book VI Introduction - Proposition VI.22 - Proposition VI.24.