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

Abstract

In (6), we associated with a given triangle $A₁A₂A₃$ 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 $d₁, d₂, d₃$ to the sides of the triangle $A₁A₂A₃$ are related by ${l₁ \over d_{1}⁰}{d_{1}²} + {l₂ \over d_{2}⁰}{d_{2}²} + {l₃ \over d_{3}⁰}{d_{3}²} = s$ (s variable in $Re$),where $l₁, l₂, l₃$ are the lengths of the sides of the triangle and where $d_{1}⁰, d_{2}⁰, d_{3}⁰$ are normal coordinates of P relative to the triangle $A₁A₂A₃$ (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 $A₁A₂A₃$ (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 $A₁A₂A₃$ and look again for the axes, especially in the cases where P = H, K, I, Z, or O.