Enumerative triangle geometry. I: The primary system, \(S\). (Q1414947)

In order to systematize the theory of triangle centers studied extensively by the author, he introduces three rational operations on the trilinear coordinates of a triangle which can be interpreted as join (or by duality intersection), parallelism, and orthogonality. Parallelism can be obtained by orthogonality after orthogonality. He then studies systematically the points and lines obtained by repeated application of these basic operations (called by him opera) which gives (a) a hierarchy of centers, and (b) some new interesting points with the certainty that by successive repetitions all centers reachable by these rational operations will be found. This is only a subset of what is geometrically reachable since, e.g., the incenter cannot be reached in this way. The extension of the domain of operation of these opera to include points defined by lines of irrational slopes is promised for the future.