Propellers in Affine Cayley–Klein Planes

Abstract

In the present paper the generalized Propeller theorem from planar Euclidean geometry is extended to all planar affine Cayley-Klein geometries. Since there are no equilateral triangles in affine Cayley-Klein planes (except for the Euclidean case), there is no direct extension of the Propeller theorem. In order to find the respective non-Euclidean analogues of it, we introduce the notion of Ω k -equilateral triangle. Some properties of such triangles are given, too. Finally, we prove a Propeller theorem related to isocentric triangles in all affine Cayley-Klein planes. Mathematics Subject Classification (2000). 51M05, 51A20, 51A25.

On the Moving Coordinate System and Euler-Savary Formula in Affine Cayley-Klein Planes

“…Much recent research is conducted in CK-planes, [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. There is a known (but not well-known) relationship between the plane geometries which have parabolic measure of distance: Euclidean, Galilean and Minkowskian (Lorentz) geometries.…”