Bivariate coordinate systems in triangle geometry.

In the plane of a triangle ABC, let L 1 as L 2 be a pair of nonparallel lines. A coordinate system having L 1 as x-axis and L 2 as y-axis is defined, and transformation formulas for representations of points and lines in (x, y) coordinates and barycentric coordinates are derived. The set of central lines consists of two types: even and odd, with consequences for the cases that L 1 and L 2 are both even, or both odd, or one even and the other odd. A mapping T(P, U) is defined for pairs (P, U) of points. This mapping maps lines to lines. A distinctive example of a bivariate coordinate system is introduced as the Steiner system, in which the axes are those of the Steiner circumellipse. In this system, the line y = x is the Euler line of ABC. [ABSTRACT FROM AUTHOR]