A relative of ``Napoleon's theorem'' (Q2721663)

The author proves the following relative of Napoleon's theorem: Given an arbitrary triangle \(ABC\) in the Euclidean plane, let \(AC'B\), \(BA'C\) and \(CB'A\) denote equilateral triangles externally erected on the sides of \(ABC\), and \(C_1, A_1, B_1\), is written for the midpoints of \(A'B'\), \(B'C'\) and \(C'A'\), respectively. Then NEWLINENEWLINENEWLINE(i) \(A_1B_1 C\), \(B_1C_1A\) and \(C_1A_1B\) are equilateral triangles with the same orientation as \(ABC\),NEWLINENEWLINENEWLINE(ii) the centroids \(A^*\), \(B^*\), \(C^*\) of these triangles form an equilateral triangle having the opposite orientation.NEWLINENEWLINENEWLINEAn analogous statement for internally erected triangles is derived, too. Finally, further relatives of Napoleon's theorem are discussed.