To construct a pyramid (regular tetrahedron), 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.
Proof
Inscribe an equilateral triangle (IV.2); erect an apex above the
centroid at height r2/3 where r is the circumradius.
The four equal edges form the tetrahedron; place the sphere through
its four vertices. The diameter-squared / side-squared =3/2.