Systematical investigation of the triangle geometry in an isotropic plane (CROSBI ID 580197)
Prilog sa skupa u zborniku | sažetak izlaganja sa skupa | međunarodna recenzija
Podaci o odgovornosti
Volenec, Vladimir
engleski
Systematical investigation of the triangle geometry in an isotropic plane
In this lecture we survey the present state of investigation of triangle geometry in an isotropic plane. The isotropic plane is a projective-metric plane, where the absolute consists of a straight line (the absolute line) and a point (the absolute point) on this line. In an isotropic plane we have the principle of duality. We shall say that an isotropic plane is an affine plane with one pointed direction, the direction of the absolute point. Ordinarily, we choose that direction for the direction of y-axis of affine coordinate system. Any point on is an isotropic point and any straight line through is an isotropic line. Any two straight lines with the same isotropic point are parallel lines and any two points on the same isotropic line are parallel points. A triangle is said to be allowable if any two of its vertices are not parallel. We shall investigate only such triangles in the isotropic plane. Conics, which touch the absolute line at the absolute point , are circles. Any allowable triangle ABC has the unique circumscibed circle. By a suitable transformation of coordinate systems we can obtain that this circle has the equation y=x2 and that the vertices of the triangle ABC are of the form A=(a, a2), B=(b, b2), C=(c, c2), where a+b+c=0. In this case we say that the triangle ABC is in standard position or that we have the standard triangle ABC In order to prove any statement on any allowable triangle it is sufficient to prove the considered statement for the standard triangle. Using the method of standard triangle and using two symmetric functions p=abc and q=bc+ca+ab of the coordinates a, b, c the systematical investigation of characteristic points, lines, circles and conics of any allowable triangle is now possible. In this lecture we shall see the following characteristic points, lines, circles and conics of the standard triangle: centroid, Feuerbach point, dual Feuerbach point, Steiner point, Gergonne point, symmedian center ; Euler line, orthic axis, Feuerbach line, Longchamps line, Lemoine line, Brocard diameter, Steiner axis ; circumcircle, incircle, Euler circle, dual Euler circle, polar circle, orthocentroidal circle, first Lemoine circle, second Lemoine circle, Apollonian circles, Thébault's circles ; circumscribed Steiner's ellipse, inscribed Steiner's ellipse, Kiepert hyperbola, Jeřábek hyperbola. Some notions and statements from Euclidean geometry have and some others have not the analogous notions and statements in the isotropic plane.
isotropic plane; standard triangle
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nije evidentirano
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nije evidentirano
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Podaci o prilogu
28-29.
2011.
objavljeno
Podaci o matičnoj publikaciji
Conference on Geometry-Theory and Applications, Abstracts
Juttler, B. ; Roschel, O.
Podaci o skupu
Conference on Geometry-Theory and Applications
pozvano predavanje
20.06.2011-24.06.2011
Vorau, Austrija