New Fantastic Curves Discovered from Rectangular Hyperbola

Abstract

By means of constructing on the hyperbola xy = N a triangle whose shape is changed under the rule that one vertex is fixed at the vertex of the hyperbola, one vertex is moving on the hyperbola and the two laterals with respect to the fixed and the moving vertices respectively keep their directions unchanged, it is discovered that the loci of the triangle's centroid and orthocenter are respectively a hyperbola and a line, the locus of the circumcenter is a new cubic algebraic curve, and those of the incenter and ex-centers are planar curves that have not been reported before. All the loci of the centers form a fantastic graph like a ying insect. Meanwhile, the discovered hyperbola and curves are merely N-dependant and can be used to estimate the distribution of y divided by x with respect to xy = N.