Teachers of elementary Mathematics are generally very glad to find some unfamiliar field into which they can turn their pupils to exercise themselves in elementary processes. The following is a suggestion of a method of approaching certain theorems in connection with the Geometry of the Triangle which, although well known to Mathematicians, are not, as a rule, studied as part of a school course, except perhaps by specialists. The Brocard Points and Circle are generally approached, via the theory of Isogonal Conjugates (see Casey’s Sequel to Euclid, Supplementary Chapter, Section I. etc.). But many of the properties connected with these points can be obtained in a more direct manner, and this suggests some interesting investigations involving nothing but the ordinary processes of Geometry and Trigonometry.