Let there be applied to BC the parallelogram BE equal to the triangle ABC [[I. 44](elem.1.44)], and to CE the parallelogram CM equal to D in the angle FCE which is equal to the angle CBL . [[I. 45](elem.1.45)]

Therefore BC is in a straight line with CF , and LE with EM .

Now let GH be taken a mean proportional to BC , CF [[VI. 13](elem.6.13)], and on GH let KGH be described similar and similarly situated to ABC . [[VI. 18](elem.6.18)]

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.](elem.6.19.p.1)] therefore, as BC is to CF , so is the triangle ABC to the triangle KGH .

But, as BC is to CF , so also is the parallelogram BE to the parallelogram EF . [[VI. 1](elem.6.1)]

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](elem.5.16)]

But the triangle ABC is equal to the parallelogram BE ; therefore the triangle KGH is also equal to the parallelogram EF .

But the parallelogram EF is equal to D ; therefore KGH is also equal to D .

And KGH is also similar to ABC .

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 .