2014/12/24 by Ruben Vigara, Vigara, Ruben
Mathematics · #51M09 #FOS: Mathematics #Metric Geometry (math.MG) #math.MG #msc:51M09
paper · pdf · doi:10.48550/arxiv.1412.7589
Comments to version 2: some minor changes and corrected typos introduced
arxiv created 2015/01/22 · arxiv updated 2015/01/23
Some translations into non-euclidean geometry of classical theorems of planar projective geometry are explored. The existence of some common triangle centers is dedeuced from theorems of Pascal and Chasles. Desargues' Theorem allows to construct a non-euclidean version of the Euler line and the nine-point circle of a triangle. The whole non-euclidean trigonometry (for triangles and generalizaed triangles) can be deduced from Menelaus' Theorem. A theorem of Carnot about affine triangles implies the non-euclidean versions of another theorem of Carnot about euclidean triangles. Finally, it is given the projective theorem which is hidden behind all the laws of cosines of non-euclidean triangles and generalized triangles.