Exparabolas of a Triangle

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.

💡 Research Summary

The paper introduces the notion of an “exparabola,” a parabola that is tangent to the three sides of a given triangle ABC and also tangent to the line at infinity. While infinitely many such parabolas exist for a triangle, the authors focus on those whose curvature‑radius (the parameter ρ) is locally maximal; these are called “max‑exparabolas.”

Using a quadratic Bézier representation of an exparabola, the authors express the squared parameter ρ² as a rational function of a single real parameter t that determines the points of tangency on the sides. Differentiating ρ² with respect to t and setting the derivative to zero yields a cubic equation
\