Characterizations by normal coordinates of special points and conics of a triangle

Abstract

In (6), we associated with a given triangle and with each point P of the Euclidean plane a pencil of homothethic ellipses or hyperbolas with center P, which are determined by the loci of the points of the plane for which the distances to the sides of the triangle are related by(Error rendering LaTeX formula) (s variable in ),where are the lengths of the sides of the triangle and where(Error rendering LaTeX formula) are normal coordinates of P relative to the triangle (see section 1). In particular, a construction for the axes of these conics is given. Several special cases are treated, where P is the orthocenter H, the Lemoine point K, the incenter I, the centroid Z, and the circumcenter O of (for a summary of these results, see section 2). In the present paper, we construct another pencil of conics with center P, using again normal coordinates relative to and look again for the axes, especially in the cases where P = H, K, I, Z, or O.