Curves of Brocard Points in Triangle Pencils in Isotropic Plane

Abstract

In this paper we consider a triangle pencil in an isotropic plane consisting of the triangles that have the same circumscribed circle. We study the locus of their Brocard points, two curves of order 4.

Loci of centers in pencils of triangles in the isotropic plane

“…It was shown in [6] that for every triangle in the isotropic plane there exist the first and second Brocard point, and they are unique. The first Brocard point B 1 is defined as the point such that its connections with the vertices A, B, C form equal angles with the sides AB, BC, and CA, respectively.…”

“…In [7] the authors considered a triangle pencil in an isotropic plane consisting of the triangles that have the same circumcircle. They studied the loci of their centroids, Gergonne points and symmedian points, while in [6] the loci of the first and second Brocards points where observed.…”

Locus Curves in Triangle Families

“…In the same way, we can observe what will be the locus of a certain line, or other object associated to triangles in a triangle family. Some results in this area, especially for the Euclidean plane can be found in [1,2,4,9,10,12], while [5,6,7] deal with the situation in the isotropic plane. This paper contains a family of triangles whose basic elements are dual to the family of triangles in [4] but the resulting locus curves are different.…”