Two generalizations of Napoleon's theorem in finite planes

AbstractThe 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