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.