×

Two generalizations of Napoleon’s theorem in finite planes. (English) Zbl 0938.51014

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 F. Bachmann and E. Schmidt [‘\(n\)-Ecke’, Mannheim-Wien-Zürich (1970; Zbl 0208.23901)] as well as J. C. Fisher, D. Ruoff and 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.

MSC:

51M04 Elementary problems in Euclidean geometries
PDFBibTeX XMLCite
Full Text: DOI