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arXiv · 2511.22347

Exparabolas of a Triangle

Martin Lukarevski·Hans-Peter Schröcker
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Abstract

Among a triangle's exparabolas (parabolas escribed to the triangle), three are distinguished by having locally maximal parameter. They are determined by a simple cubic equation and characterized by having axes that contain the triangle's centroid. More generally, there are three (not necessarily real) exparabolas with axes through a given point $X$. Their focal points determine another triangle which we call the $X$-focal triangle. It shares the circumcircle with the original triangle and its orthocenter is $X$. The sequence of iterated focal triangles with respect to the centroids splits into an even and an odd sub-sequence that both converge to equilateral triangles.

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Martin Lukarevski, Hans-Peter Schröcker. 2025-11-27. Exparabolas of a Triangle. https://doi.org/10.1007/s00022-026-00798-5

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