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


Abstract


In (6), we associated with a given triangle A<sub>1</sub>A<sub>2</sub>A<sub>3</sub> 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 d<sub>1</sub>, d<sub>2</sub>, d<sub>3</sub> to the sides of the triangle A<sub>1</sub>A<sub>2</sub>A<sub>3</sub> are related by(Error rendering LaTeX formula) (s variable in Re),where l<sub>1</sub>, l<sub>2</sub>, l<sub>3</sub> are the lengths of the sides of the triangle and where(Error rendering LaTeX formula) are normal coordinates of P relative to the triangle A<sub>1</sub>A<sub>2</sub>A<sub>3</sub> (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 A<sub>1</sub>A<sub>2</sub>A<sub>3</sub> (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 A<sub>1</sub>A<sub>2</sub>A<sub>3</sub> and look again for the axes, especially in the cases where P = H, K, I, Z, or O.

DOI Code: 10.1285/i15900932v24n1p9

Keywords: Euclidean plane; Triangle center; Trilinear coordinates

Classification: 51N20; 51M04

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