and for this reason again the semicircle described on GK will also pass through F. Similarly it will also pass through the remaining angular points of the cube. If then, KG remaining fixed, the semicircle be carried round and restored to the same position from which it began to be moved, the cube will be comprehended in a sphere. I say next that it is also comprehended in the given sphere. For, since GF is equal to FE, and the angle at F is right, therefore the square on EG is double of the square on EF. But EF is equal to EK; therefore the square on EG is double of the square on EK; hence the squares on GE, EK, that is the square on GK [I. 47], is triple of the square on EK. And, since AB is triple of BC, while, as AB is to BC, so is the square on AB to the square on BD, therefore the square on AB is triple of the square on BD. But the square on GK was also proved triple of the square on KE. And KE was made equal to DB; therefore KG is also equal to AB. And AB is the diameter of the given sphere; therefore KG is also equal to the diameter of the given sphere. Therefore the cube has been comprehended in the given sphere; and it has been demonstrated at the same time that the square on the diameter of the sphere is triple of the square on the side of the cube. Q. E. D.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
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