KoG, Vol. 22 No. 22, 2018.
Original scientific paper
https://doi.org/10.31896/k.22.1
Two Convergent Triangle Tunnels
Boris Odehnal
orcid.org/0000-0002-7265-5132
; University of Applied Arts Vienna, University of Applied Arts Vienna, University of Applied Arts Vienna, Vienna, Austria
Abstract
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.
Keywords
triangle; semi-orthogonal path; Brocard points; symmedian point; discrete logarithmic spiral; Tucker-Brocard cubic
Hrčak ID:
214640
URI
Publication date:
3.1.2019.
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