Symmedian

Abstract: Given a point and a rational curve in the plane, their bisector curve is rational [Farouki and Johnston 1994a]. However, in general, the bisector of two rational curves in the plane is not rational [Farouki and Johnstone 1994b]. Given a point and a rational space curve, this art icle shows that the bisector surface is a rational ruled surface. Moreover, given two rational space curves, we show that the bisector surface is rational (except for the degenerate case in which the two curves are coplanar).

Abstract: In the flourishing days of triangle geometry, many special points were discovered and investigated. Apart from well-known points like the centroid (or median point), the orthocenter, and the circumcenter, 'new' triangle points were studied, points called the symmedian point (or point of Lemoine) and the points of Gergonne, Nagel, Torelli, and Brocard, to name but a few. There is little doubt in my mind that the Brocard points rank amongst the most interesting of these special points associated with the triangle. Although general interest has long since waned and results once regarded as important have sunk into oblivion, it might still be worth our while to revive some of the gems of triangle geometry. In the brilliant light of modern knowledge we might even discover new and interesting insights. In the literature on Euclidean geometry some books can be singled out that deal exclusively with the geometry of the triangle and the circle. An excellent monograph is [4], and for those with a smattering of German [3] gives much information, too; [5] is of a more general nature, but this work also contains many pages devoted to the triangle and its associated points. Finally, the Brocard configuration is the singular topic of Emmerich's treatise [2], recommendable for its proverbial 'Griindlichkeit.' In the rich field of Brocardian geometry, our attention shall be focused on the set of triangles equibrocardal to a given triangle (T). In order to explain the terminology, our first concern should be with the reader who wishes to be introduced to the Brocard points and the Brocard angle of a plane triangle. Well then, given a triangle (T) with vertices A1, A2, and A3, notation: (T) = A1A2A3, the first (or positive) Brocard point of (T) is the unique point Q such that the angles ZQA1A2, zQA2A3, and zQA3A1 are equal. The second (or negative)