The Euler line of a triangle is mostly valued, not for any practical application, but purely as a beautiful, esoteric example of post-Greek geometry. Much to his surprise, however, the author recently came across the following result and theorem by Sylvester (1814-1897) in [1] that involves an interesting application of forces that relate to the Euler line (segment). This result is also mentioned in [2] without proof or reference to Sylvester.

Every tetrahedron has a set of quadritangent spheres, that is, spheres that touch the four planes containing its faces. In this article I discuss three questions about them.

1. How many quadritangent spheres does a given tetrahedron have?

2. How can we find their sizes and positions with as little calculation as possible?

3. What relations exist between the vertices and points of contact that lie on anyone of the planes?

In a previous note [1], Fermat's method of descente infinie was used to prove that the equations.

have no positive integer solutions. The geometrically based proof of [1] masked the underlying use of the difference of two squares. In the proofs of this article we shall make its use explicit, just as Fermat did [2, pp. 293-294].

We shall use the elementary idea of the difference of two squares to develop a powerful technique for solving equations of the form ax4 + bx2y2 + cy4 = z2. This will then be applied to three problems of historical interest.

We elaborate the idea behind Markov chain Monte Carlo (MCMC) methods in a mathematically coherent, yet simple and understandable way. To this end, we prove a pivotal convergence theorem for finite Markov chains and a minimal version of the Perron-Frobenius theorem. Subsequently, we briefly discuss two fundamental MCMC methods, the Gibbs and Metropolis-Hastings sampler. Only very basic knowledge about matrices, convergence of real sequences and probability theory is required.