New features of Napoleon's triangles (Q1818896)

For a given triangle ABC, the author obtains some possibly new statements on the (generalized) Torricelli figure, which is originally formed by three equilateral triangles erected on the sides of ABC and therefore has a strong relation to Napoleon's theorem. For example, he considers segments and triangles generated by further points in this configuration (e.g., by the midpoints of the segments joining the exterior points of the erected triangles and the midpoints of the sides of ABC, or by the centroids and circumcenters of the erected triangles) if the erected triangles are not necessarily equilateral. His main aim is to deduce elementary properties of these point configurations from certain assumptions on the shape of the erected triangles.