Some triangle geometry of the triangle family

In this work we consider a one-parameter triangle family T. We prove that the set of the orthocenter, centroid, circumcenter, and some other sets of the triangle centers lie on different hyperbolae. Furthermore, it will be shown that the hyperbolae which are the sets of triangle centers that lie on the Euler lines of the triangle family T have a space interpretation with the Hohenberg transformation. In this presentation we also prove and construct two different cubics as the line envelopes of the side bisectors and Euler lines, and two different quartics as the circle envelopes of the circumcircles and Euler circles of the triangle family T.