FORUM GEOM ISSN 1534-1178 The Lemoine Cubic and Its Generalizations

Abstract. For a given triangle, the Lemoine cubic is the locus of points whose cevian lines intersect the perpendicular bisectors of the corresponding sides of the triangle in three collinear points. We give some interesting geometric prop-erties of the Lemoine cubic, and study a number of cubics related to it. 1. The Lemoine cubic and its constructions In 1891, Lemoine published a note [5] in which he very briefly studied a cubic curve defined as follows. Let M be a point in the plane of triangle ABC. Denote by Ma the intersection of the line MA with the perpendicular bisector of BC and define Mb and Mc similarly. The locus of M such that the three points Ma, Mb, Mc are collinear on a line LM is the cubic in question. We shall denote this cubic by K(O), and follow Neuberg [8] in referring to it as the Lemoine cubic. Lemoine claimed that the circumcenter O of the reference triangle was a triple point of K(O). As pointed out in [7], this statement is false. The present paper considerably develops and generalizes Lemoine’s note. We use homogeneous barycentric coordinates, and adopt the notations of [4] for triangle centers. Since the second and third coordinates can be obtained from the first by cyclic permutations of a, b, c, we shall simply give the first coordinates. For convenience, we shall also writ