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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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