Triangle family and Euler

In this work we consider a one-parameter triangle family T, determined by two arbitrary lines and a pencil, whose vertex does not lie on these two chosen lines. It will be shown and proved that the known points of Euler line, in this case the set of the orthocenter, centroid, circumcenter and nine-point center lie on different hyperbolae. Furthermore, in this presentation we prove and show some properties of two different curves: the line envelope of Euler lines and the circle envelope of the Euler’s circles of the triangle family T. Also there will be discussed and constructed some other line envelopes that can be related to the triangle family T and some points of the Euler line or Euler’s circle.