Book 6
25
Then, since, as BC is to GH, so is GH to CF, and, if three straight lines be proportional, as the first is to the third, so is the figure on the first to the similar and similarly situated figure described on the second, [VI. 19, Por.] therefore, as BC is to CF, so is the triangle ABC to the triangle KGH.
25
But, as BC is to CF, so also is the parallelogram BE to the parallelogram EF. [VI. 1]
25
Therefore also, as the triangle ABC is to the triangle KGH, so is the parallelogram BE to the parallelogram EF; therefore, alternately, as the triangle ABC is to the parallelogram BE, so is the triangle KGH to the parallelogram EF. [V. 16]
25
But the triangle ABC is equal to the parallelogram BE; therefore the triangle KGH is also equal to the parallelogram EF.
25
But the parallelogram EF is equal to D; therefore KGH is also equal to D.
25
And KGH is also similar to ABC.
25
Therefore one and the same figure KGH has been constructed similar to the given rectilineal figure ABC and equal to the other given figure D. Q. E. D. to which the figure to be constructed must be similar, literally to which it is required to construct (one) similar, ᾧ δεῖ ὅμοιον συστήσασθαι.
PROPOSITION 26.
26
If from a parallelogram there be taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, it is about the same diameter with the whole
