Conics and Transformations Defined by the Parallelians of a Triangle

For any point P in the Euclidean plane of a triangle Δ, the six parallelians of P lie on a single conic, which shall be called the parallelian conic of P with respect to Δ. We provide a synthetic and an analytic proof of this fact. Then, we studied the shape of this particular conic, depending on the choice of the pivot point P. This led to the finding that the only circular parallelian conic is the first Lemoine circle. Points on the Steiner inellipse produce parabolae, and those on a certain central line yield equilateral hyperbolae. The hexagon built by the parallelians has an inconic I and the tangents of P at the parallelians define some triangles and hexagons with several circum- and inconics. Certain pairings of conics, together with in- and circumscribed polygons, give rise to different kinds of porisms. Further, the inconics and circumconics of the triangles and hexagons span exponential pencils of conics in which any pair of subsequent conics defines a new conic as the polar image of the inconic with regard to the circumconic. This allows us to construct chains of nested porisms. The trilinear representations of the centers of the appearing conics, as well as the perspectors of some deduced triangles, depending on the indeterminate coordinates of P, define some algebraic transformations that establish algebraic relations between well- and lesser-known triangle centers. We completed our studies by compiling a list of possible porisms between any pair of conics. Further, we describe the possible loci of pivot points so that the mentioned conics allow for porisms of polygons with arbitrary numbers of vertices.