Kitap 13
12
Then, since the triangle ABC is equilateral, therefore the circumference BEC is a third part of the circumference of the circle ABC.
12
Therefore the circumference BE is a sixth part of the circumference of the circle; therefore the straight line BE belongs to a hexagon; therefore it is equal to the radius DE. [IV. 15, Por.]
12
And, since AE is double of DE, the square on AE is quadruple of the square on ED, that is, of the square on BE.
12
But the square on AE is equal to the squares on AB, BE; [III. 31, I. 47] therefore the squares on AB, BE are quadruple of the square on BE.
12
Therefore, separando, the square on AB is triple of the square on BE.
12
But BE is equal to DE; therefore the square on AB is triple of the square on DE.
12
Therefore the square on the side of the triangle is triple of the square on the radius. Q. E. D.
PROPOSITION 13.
13
To construct a pyramid, to comprehend it in a given sphere, and to prove that the square on the diameter of the sphere is one and a half times the square on the side of the pyramid.
