Abstract
There are three affine Cayley-Klein planes (see [5]), namely, the Euclidean plane, the isotropic (Galilean) plane, and the pseudo-Euclidean (Minkow-skian or Lorentzian) plane. We extend the generalization of the well-known Napoleon theorem related to similar triangles erected on the sides of an arbitrary triangle in the Euclidean plane to all affine Cayley-Klein planes. Using the Ωk-and anti-Ωk-equilateral triangles introduced in [28], we construct the Napoleon and the Torricelli triangle of an arbitrary triangle in any affine Cayley-Klein plane. Some interesting geometric properties of these triangles are derived.
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