Abstract.
Let (A 2,\( \mathcal{C} \)) be a Minkowski plane with a centrally symmetric, strictly convex C 1-curve \( \mathcal{C} \) as the unit circle. Then \( \mathcal{C} \) induces in (A 2,\( \mathcal{C} \)) a left-orthogonality structure '\( \dashv \)' by setting tangents of \( \mathcal{C} \) (and their parallels) left-orthogonal to the corresponding radii (and their paralles). If a line g is left-orthogonal to another one h, then h is right-orthogonal to \( g, (h \vdash g) \). Based on those concepts of orthogonality in (A 2, \( \mathcal{C} \)) left- and right-altitudes of a triangle are defined and one can discuss the existence of left- or right-orthocentric triangles. In general Minkowski planes these concepts of orthocenters are independent of a third type of a triangle-orthocenter, which is based on a circle-geometric definition due to Asplund and Grünbaum, c.f. [1].¶¶Further results are the following: In every plane A 2,\( \mathcal{C} \) there exist triplets of directions \( \overline{g}_i \) such that the triangles \( \mathcal {T} \) having sides g i parallel to \( \overline{g}_i \) are left-orthocentric. A plane A 2,\( \mathcal{C} \) is euclidean, iff each triangle \( \mathcal {T} \) is left-orthocentric. Constructing the altitudes of an altitude-triangle of a non (left- or right-)-orthocentric triangle \( \mathcal {T} \) starts iteration processes with attractors (resp. repulsors) which can be called ‘limit orthocenters’ to the given triangle \( \mathcal {T} \).
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Received 27 April 1999; revised 16 January 2002.
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Weiss, G. The concepts of triangle orthocenters in Minkowski planes. J.Geom. 74, 145–156 (2002). https://doi.org/10.1007/PL00012533
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DOI: https://doi.org/10.1007/PL00012533