On some Curves Related to Pencil of Triangles in Isotropic Plane

We observe a pencil of triangles in isotropic plane whose vertices A and B are fixed, while vertex $C$ moves on the circumscribed circle. We prove that the locus of all centroids of triangles in such a pencil is a circle, the locus of all Gergonne points is a curve of order 4 and the locus of all symmedian centers is an ellipse. For the tangential triangles of all the triangles in such a pencil we prove that the locus of all centroids of all triangles is a line, the locus of all Gergonne points is an ellipse and the locus of all symmedian centers is a curve of order 3.