Charles Thas,  A  Generalization of the Lemoine Point,

Forum Geometricorum, 3 (2003) 161--167.

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 d_1, d_2, and d_3 to the sides of the triangle takes its minimal value. There are several ways to generalize the Lemoine point. First, we can consider n \ge 3 lines u_1, ..., u_n instead of three in the Euclidean plane and search for the point which minimalizes the expression d_1^2 + ... + d^2_n, where d_i is the distance to the line u_i, i=1, ..., n. Second, we can work in the Euclidean m-space R^m and consider n hyperplanes in R^m with n \ge m+1. In this paper a combination of these two generalizations is presented.

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