Regular hexagons associated to centroid sharing triangles (Q1267322)

Let \(s\) and \(t\) be real numbers different from \(-1\) such that \(st+s+1\neq 0\) and \(st+t+1\neq 0\), and denote by \(\mathbb{R}^2\) the real Euclidean plane. For points \(U,V\in\mathbb{R}^2\) and any real number \(r\neq-1\) let \(r(U,V)\) be the point \(U\) when \(U=V\) and the unique point \(T\) on the line \(UV\) such that \(\overline{UT}/\overline{TV}=r\) (affine ratio) when \(U\neq V\). For any (non-degenerate) triangle \(UVW\), the point of intersection of the two lines \(Us(V, W)\), \(Vt(W,U)\) is well-determined and will (here) be denoted by \((UVW)\). Using complex numbers, the author proves the Theorem (abridged): Let \(ABC\), \(PQR\), and \(XYZ\) be triangles whose centroids coincide such that neither \(A,P,X\) nor \(B, Q,Y\) nor \(C,R,Z\) are collinear. Let \(P_1=(APX)=P_7\), \(P_2=(AXP)\), \(P_3=(BQY) \), \(P_4=(BYQ)\), \(P_5=(CRZ)\), \(P_6=(CZR)\). For \(i=1,\dots,6\), let \(Q_iP_iP_{i+1}\) be the equilateral triangles built on the sides of the hexagon \(P_1P_2,\dots, P_6\) all with the same orientation. Let \(F_i\) be the centroid of the triangle \(Q_{i-1}Q_iQ_{i+1}\), \(i=1,\dots,6\). Let \(R_iQ_{i-1}Q_{i+1}\), \(i=1,\dots,6\), be the equilateral triangles built on the small diagonals of the hexagon \(Q_1Q_2 \dots Q_6\) all with the same orientation as equilateral triangles used in the construction of points \(Q_1,Q_2,\dots,Q_6\). Let \(R_0=R_6\) and \(R_7=R_1\). Let \(G_i\) be the centroid of the triangle \(R_{i-1}R_iR_{i+1}\), \(i=1,\dots,6\). Then \(F_1F_2\dots F_6\) and \(G_1G_2\dots G_6\) are regular hexagons whose centers coincide with the centroid of the triangle \(ABC\).