Functional equations associated with triangle geometry

Summary Triangle geometry is treated in the context of functional equations of three variablesa, b, c which may be regarded as the sidelengths of a variable triangle. Trianglecenters (e.g., incenter, circumcenter, centroid), andcentral lines (e.g., the Euler line) are defined and partitioned into classes:0-centers, 1-centers, 2-centers and0-lines, 1-lines, and 2-lines. Criteria for parallelism, perpendicularity, and other geometric relations are proved in terms of these classes. The Euler line and central lines parallel or perpendicular to the Euler line serve as examples.