Explorations of the Brocard Point

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One of the surprising characteristics of triangles is the number of points that lie at the intersection of three, similarly defined lines, segments, or rays. The centroid of a triangle, for instance, is the intersection of the three medians; the in-center is the intersection of the three angle bisectors; the circumcenter is the intersection of the perpendicular bisectors of the three sides; and the orthocenter is the intersection of the three altitudes. Defined in terms of common geometric objects and being relatively easy to construct, these points of concurrency have been studied by generations of secondary students. In this article, we describe yet another point of concurrency within a triangle: the Brocard point. Unlike the prior points, which have been known to mathematicians for millennia, the Brocard point is a relatively recent discovery. Like the prior points, however, the Brocard point is easily accessible to secondary students and can serve as a rich context for explorations.