Two generalizations of Napoleon's theorem in finite planes (Q1808806)

The theorem of Napoleon says that the triangle, whose vertices are the circumcenters of the equilateral triangles all erected externally (or all internally) on the sides of an arbitrarily given triangle, is equilateral. Based on an algebraic method due to \textit{F. Bachmann} and \textit{E. Schmidt} [`\(n\)-Ecke', Mannheim-Wien-Zürich (1970; Zbl 0208.23901)] as well as \textit{J. C. Fisher}, \textit{D. Ruoff} and \textit{J. Shilleto} [``Polygons and polynomials'', in: The Geometric Vein, Springer, New York, 321-333 (1981; Zbl 0497.51018)], the author proves two generalizations of this theorem in Galois planes of odd order. In the spirit of related extensions of Napoleon's theorem obtained by Barlotti and Neumann, these generalizations refer to a given \(n\)-gon \(P\) and further \(n\)-gons, corresponding to \(P\) by analogous erection procedures or similar relations and again yielding a final \(n\)-gon.