Construction of Polyhedron in Outer of Concentric Spheres/Porism

Proof

 * Euclid-XII-17.png

We have that the polyhedron as constructed in is divided into pyramids which are similar in number and arrangement.

But from :
 * similar pyramids are to one another in the triplicate ratio of their corresponding sides.

Therefore the pyramid whose base is $KBPS$ and whose apex is $A$ has to the similarly arranged pyramid in the other sphere the triplicate ratio of their corresponding sides: that is, $AB$ to the radius of the other sphere.

The same applies to all the other pyramids.

It follows from :
 * the sum total of all the pyramids that form the whole polyhedron in the one sphere has to the sum total of all the pyramids that form the whole polyhedron in the other sphere the triplicate ratio of the radii of the spheres.

That is, the triplicate ratio of the diameters of the spheres.