A Special Family of Triangles

We study the one-parameter family of triangles that emerges if one sideline traces a pencil of lines and the opposite angle is fixed. A description of the traces of triangle centers and the pair of Brocard points in terms of parametrizations and equations is given. The envelopes of the families of circumcircles and nine-point circles are determined. Our approach even allows us to consider and treat the triangle family as a two-parameter family of triangles; i.e., the (interior) angle opposite to the pencil, which is in the beginning fixed, may also change.