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.

In this paper, we collect some results on a problem which has been considered by many authors over the years, including Ed Pegg Jr., R. M. Kurchan, William Rex Marshall, David Neary, Richard Korf and Eric Huang. The results have been rediscovered many times by many people, including the present authors, but very little has been published, except in a little known journal which no longer appears. Some of the cases have been solved by hand while others have been covered by Korf [1, 2], using a computer. We see our main role as that of collectors of information and presenters of results, in the hope that others will be inspired to study what is an exciting and very interesting area of combinatoric geometry; we believe, however, that some of our observations, variations and upper bounds are new.

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?

Dedicated to the memory of Russell Towle, a remarkable man who contributed so much to geometry and to other aspects of the quality of life.

We introduce an expanded notation where r + s = n, for the binomial coefficient , and then use this expanded notation to develop theorems involving 8 binomial coefficients, analogous to the Star of David Theorem, which. in its original form, involved the 6 neighbours of a given binomial coefficient in the Pascal Triangle (see Section 3), that appeared in [1,2,3,4,5,6,7,8,9].

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.