Napoleon's theorem and generalizations through linear maps (Q1856592)

Let \(a_1a_2a_3\) be a triangle in \(\mathbb{E}^2\) all sides of which are suitably divided into three parts. More precisely, for given \(\lambda, \overline \lambda\in \mathbb{R}\) define two point triples \(b_1,b_2,b_3\) and \(\overline b_1,\overline b_2,\overline b_3\) as affine combinations \(b_i=\lambda a_i+ (1-\lambda) a_{i+1}\) \((i=1,2,3\); indices modulo 3) and \(\overline b_i\) analogously with the help of \(\overline\lambda\). Based on equilateral triangles erected on the sides \(b_1\overline b_1\), \(\overline b_1b_2, \dots, \overline b_3 b_1\) of the hexagon \(b_1\overline b_1b_2\overline b_2b_3\overline b_3\) one can construct a hexagonal extension of the well known Napoleon figure. Continuing corresponding constructions and observations of J. Fukuta and Z. Čerin, the author shows that these results are closely related to linear maps and finds with this approach also new theorems.