Book 6
22
Since then, as AB is to CD, so is EF to QR, and there have been described on AB, CD the similar and similarly situated figures KAB, LCD, and on EF, QR the similar and similarly situated figures MF, SR, therefore, as KAB is to LCD, so is MF to SR.
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But also, by hypothesis, as KAB is to LCD, so is MF to NH; therefore also, as MF is to SR, so is MF to NH. [V. 11]
22
Therefore MF has the same ratio to each of the figures NH, SR; therefore NH is equal to SR. [V. 9]
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But it is also similar and similarly situated to it; therefore GH is equal to QR.
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And, since, as AB is to CD, so is EF to QR, while QR is equal to GH, therefore, as AB is to CD, so is EF to GH.
22
Therefore etc. Q. E. D.
PROPOSITION 23.
23
Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.
23
Let AC, 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.
23
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.
23
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]
23
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.
