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

  175-181

  Zoltán Szilasi

  Views:

  123

  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.

• Ein ungewöhnlicher Weg zu Jakob Steiners Umellipse eines Dreiecks und zur Steiner–Hypozykloide

  49-65

  Oswald Giering

  Views:

  71

  In real projective geometry of triangles two problems of collinear points are discussed. The problems differ only from the running through the vertices of a given triangle ABC. Resolving the problems we find two cubic curves kS and kT . Affine specialization leads to the circumscribed Steiner ellipse about the triangle ABC and shows us this ellipse in more general surroundings. Euclidean specialization leads to Steiners three-cusped hypocycloid.

• On the nine-point conic of hyperbolic triangles

  195-211

  Zoltán Szilasi

  Views:

  44

  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