A Generalization of the Lemoine Point

Abstract. It is known that the Lemoine point K of a triangle in the Euclidean plane is the point of the plane where the sum of the squares of the distances d1, d2, and d3 to the sides of the triangle takes its minimal value. There are several ways to generalize the Lemoine point. First, we can consider n ≥ 3 lines u1,..., un instead of three in the Euclidean plane and search for the point which minimalizes the expression d21 + · · ·+d2n, where di is the distance to the line ui, i = 1,..., n. Second, we can work in the Euclidean m-space Rm and consider n hyperplanes in Rm with n ≥ m + 1. In this paper a combination of these two generalizations is presented. 1