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To the isotropic generalization of Wallace lines. (English) Zbl 1006.51020
In the isotropic plane we choose a point \(X\) and a non-isotropic line \(g\); for a given non-zero real \(\delta\) there exists one line \(h\) through \(X\) such that the isotropic angle \((h,g)\) equals \(\delta\); the common point of \(g\) and \(h\) is called the \(\delta\)-footpoint of \(X\) on \(g\) [cf. J. Lang, Ber. Math.-Stat. Sekt. Forschungszentrum Graz 205, 11 S. (1983; Zbl 0516.51020)]. If \(\Delta\) is an admissible triangle and \(X\not\in\Delta\) is a point of the (isotropic) circumcircle of \(\Delta\) [cf. H. Sachs, ‘Ebene isotrope Geometrie’, Vieweg-Verlag, Wiesbaden (1987; Zbl 0625.51001)], then the triangle of \(\delta\)-footpoints of X on the sides of \(\Delta\) degenerates to the isotropic Wallace line \(\omega(X,\delta)\) of \(X\) with respect to \(\delta\). (Other notations for the Euclidian Wallace line are: pedal line or Simson line [cf. C. Kimberling, Congr. Numerantium 129, 1-285 (1998; Zbl 0912.51009)].)
Among others the author shows that (for fixed \(\delta\)) all Wallace lines of \(\Delta\) envelop a parabola of Neil. The results are expressively illustrated in two figures.
MSC:
51N25 Analytic geometry with other transformation groups
51N30 Geometry of classical groups
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