Maclaurin Trisectrices as t-Affine Loci of the First Isogonic and Isodynamic Centers

For a given non-degenerate triangle \(\triangle \) , we define the t-affine family to consist of triangles \(\triangle _t\) whose vertices are pairwise affine combinations of the vertices of \(\triangle \) , all parameterized by the same value of t. We demonstrate that the first isogonic centers \(X_{13,t}\) and the first isodynamic centers \(X_{15,t}\) of \(\triangle _t\) lie on two Maclaurin trisectrices and differ by scaling and a rotation about the barycenter of \(\triangle \) .