Two generalizations of Napoleonʹs theorem in finite planes Original Research Article

The following theorem about triangles in the Euclidean plane is attributed to Napoleon: Let A=A1A2A3 be a triangle in the Euclidean plane and B=B1B2B3 be the triangle whose vertices are the centers of the equilateral triangles all erected externally (or all internally) on the sides of A. Then B is an equilateral triangle. Two generalizations of this theorem in Galois planes of odd order are given. The proofs are based on an algebraic method which was developed by Bachmann and Schmidt (n-Ecke, Hochschultaschenbücher Verlag, Mannheim, Wein, Zurich, 1970) and Fisher et al. (The Geometric Vein, Springer, New York, 1981, pp. 321–333) to deal with geometry problems.