Two Convergent Triangle Tunnels

A semi-orthogonal path is a polygon inscribed into a given polygon such that the i-th side of the path is orthogonal to the i-th side of the given polygon. Especially in the case of triangles, the closed semi-orthogonal paths are triangles which turn out to be similar to the given triangle. The iteration of the construction of semi-orthogonal paths in triangles yields infinite sequences of nested and similar triangles. We show that these two different sequences converge towards the bicentric pair of the triangle's Brocard points. Furthermore, the relation to discrete logarithmic spirals allows us to give a very simple, elementary, and new constructions of the sequences' limits, the Brocard points. We also add some remarks on semi-orthogonal paths in non-Euclidean geometries and in n-gons.

Key words: triangle, semi-orthogonal path, Brocard points, symmedian point, discrete logarithmic spiral, Tucker-Brocard cubic

Jeřabek Hyperbola of a Triangle in an Isotropic Plane

In this paper, we examine the Jeřabek hyperbola of an allowable triangle in an isotropic plane. We investigate different ways of generating this special hyperbola and derive its equation in the case of a standard triangle in an isotropic plane. We prove that some remarkable points of a triangle in an isotropic plane lie on that hyperbola whose centre is at the Feuerbach point of a triangle. We also explore other interesting properties of this hyperbola and its connection with some other significant elements of a triangle in an isotropic plane.

Key words: allowable triangle, standard triangle, Jeřabek hyperbola

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Curves of Brocard Points in Triangle Pencils in Isotropic Plane

In this paper we consider a triangle pencil in an isotropic plane consisting of the triangles that have the same circumscribed circle. We study the locus of their Brocard points, two curves of order 4.

Key words: isotropic plane, triangle pencil, Brocard points

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The Universal Parabola

We develop classical properties, as well as some novel facts, for the parabola using the more general framework of rational trigonometry. This extends the study of this conic to general fields.

Key words: parabola, rational trigonometry, conic

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The Three Reflections Theorem Revisited

It is well-known that, in a Euclidean plane, the product of three reflections is again a reflection, iff their axes pass through a common point. For this "Three reflections Theorem" (3RT) also non-Euclidean versions exist, see e.g. [4]. This article presents affine versions of it, considering a triplet of skew reflections with axes through a common point. It turns out that the essence of all those cases of 3RT is that the three pairs (axis, reflection direction) of the given (skew) reflections can be observed as an involutoric projectivity. For the Euclidean case and its non-Euclidean counterparts this property is automatically fulfilled. From the projective geometry point of view a (skew) reflection is nothing but a harmonic homology. In the affine situation a reflection is an indirect involutoric transformation, while "direct" or "indirect" makes no sense in projective planes. A harmonic homology allows an interpretation both, as an axial reflection and as a point reflection. Nevertheless, one might study products of three harmonic homologies, which result in a harmonic homology again. Some special mutual positions of axes and centers of the given homologies lead to elations or even to the identity, too. A consequence of the presented results are further generalisations of the 3RT, e.g. in planes with Minkowski metric, affine or projective 3-space, or in circle geometries.

Key words: three reflections theorem, axial reflection, harmonic homology, involutoric projectivity

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