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• Charakteristische Dreieckpunkte in der projektiv-erweiterten hyperbolischen Ebene

  299-315

  Ana Sliepčević

  Ivanka Babić

  Views:

  56

  Some basic planimetric constructions regarding segments, angles and triangles are shown in the Cayley-Klein model of the hyperbolic plane. Relationship with the situation in the Euclidean plane is given. H-triangles are classified considering the location of their vertices and sides with respect to the absolute. There are 28 types of triangles. It is shown that there exist 12 pairs of dual triangles, while 4 types of triangles are dual to themselves. For every type of triangle the existence and number of the characteristic points are determined. Few examples of triangles with construction of their characteristic points, incircles and circumcircles are given.

• On the nine-point conic of hyperbolic triangles

  195-211

  Zoltán Szilasi

  Views:

  20

  In the Cayley–Klein model, we review some basic results concerning the geometry of hyperbolic triangles. We introduce a new definition of the circumcircle of a hyperbolic triangle, guaranteed to exist in every case, and describe its main properties. Our central theorem establishes, by means of purely elementary projective geometric arguments, that a hyperbolic triangle has a nine-point conic if and only if it is a right triangle.

  Subject Classification: 51M09

• Classical theorems on hyperbolic triangles from a projective point of view

  175-181

  Zoltán Szilasi

  Views:

  118

  Using the Cayley-Klein model of hyperbolic geometry and the tools of projective geometry, we present elementary proofs for the hyperbolic versions of some classical theorems on triangles. We show, in particular, that hyperbolic triangles have no Euler line.