| If from a parallelogram there is taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, then it is about the same diameter with the whole. | ||
| From the parallelogram ABCD let there be taken away the parallelogram AF similar and similarly situated to ABCD, and having the angle DAB common with it.
I say that ABCD is about the same diameter with AF. | ||
| For suppose it is not, but, if possible, let AHC be the diameter. Produce GF and carry it through to H. Draw HK through H parallel to either of the straight lines AD or BC. | I.31 | |
| Since, then, ABCD is about the same diameter with KG, therefore DA is to AB as GA is to AK. | VI.24 | |
| But also, since ABCD and EG are similar, therefore DA is to AB as GA is to AE. Therefore GA is to AK as GA is to AE. | VI.Def.1
V.11 | |
| Therefore GA has the same ratio to each of the straight lines AK and AE. | ||
| Therefore AE equals AK the less equals the greater, which is impossible. | V.9 | |
| Therefore ABCD cannot fail to be about the same diameter with AF. Therefore the parallelogram ABCD is about the same diameter with the parallelogram AF. | ||
| Therefore, if from a parallelogram there is taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, then it is about the same diameter with the whole. | ||
| Q.E.D. | ||
Book VI Introduction - Proposition VI.25 - Proposition VI.27.