Do you deplore “watertight compartments”? Here is a good way to ignore them.

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Although the primary purpose of this article is to exhibit and prove two interesting relationships involving the binomial coefficients that we have not encountered elsewhere,1 we have several subsidiary objectives. We endeavor to show how the relationships arose naturally in the course of solving other problems; we attempt to strip them of the usual aura of mystery that often surrounds a nontrivial mathematical result; and we encounter an aspect of mathematical induction not often mentioned to students, nor even known to many teachers of mathematics. In this way we hope to endow the article with heuristic as well as mathematical content.

The problem of determining integer-sided triangles that contain a 60-degree angle was mentioned by R. G. Dearborn (1985), In this article I shall derive some general formulas for resolving this problem; these include Dearborn's result as a special case and have several further ramifications, including a connection with the well-known “SSA ambiguity” in elementary plane geometry.

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