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Proposition 23

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 VI.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 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.

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(Forthcoming)

Book VI Introduction - Proposition VI.22 - Proposition VI.24.

© 1996
D.E.Joyce
Clark University