Isotomic Directions

The parabola inscribed to a triangle has increasingly taken on amost mysterious presence to me. On the set of conics they have zero measure, so I do not expect them to turn up often, and when they do, I assume they will not be of particular interest. We usually first hear about them in the form of the Kiepert parabola, which is sometime presented with the more famous Kiepert hyperbola.

The first thing that struck me is this: if a circumconic has perspector at infinity, there is automatically a dual inscribed parabola whose perspector is on the Steiner ellipse. Perspectors at infinity are common enough, so inparabolae are common enough too.

Wilson Stother discovered that the asymptotes of a circumconic are isotomic. This began a discussion that culminated with

[François Rideau] Every point M outside the medial triangle defines 2 lines isotomic through this point M.
[JPE] If you consider a point M lying on a parabola inscribed in the medial triangle, the isotomic line of the tangent at M to the parabola goes through M too. Hence, if I well understand what you mean, the integral curves of your flow should be the parabolae inscribed in the medial triangle.
Friendly. Jean-Pierre [Ehrman]

Jean-Pierre Ehrman pointed out that these isotomic directions were the tangents to the meeting point of two parabolas inscribed to the medial triangle. This is simply the most interesting thing I have learned recently.

Here is the picture showing the distribution of isotomic pairs of lines, which of course would indicate the directions of the asymptotes of a circumconic centered at that point. The blue areas are bounded by the edges of the medial triangle. Inside these regions there cannot be the center of a circumscribed hyperbola. The asym at a distribution of points are shown. In light grey are a number of the parabolae inscribed to the medial triangle.

Note that point on the nine point circle host perpendicular lines. The lines degenerate at the medial edges. On an edge of ABC one line is parallel to an edge. The duals of the axes on the nine point circle will lie on the Simmons (?) cubic.

For picture below: P is at the intersection of two parabolas inscribed to the medial triangle. The lines tangent to the two parabolas are isotomic lines. If P were the center of an hyperbola, the red lines would be its asymptotes. The isotomic lines are perpendicular when P is on the 9 pt circle (blue).

When P, representing the center of a rectangular parabola, crosses a sideline of the medial triangle, the perspector = dtP goes inside the inner Steiner ellipse and the conic is no longer an hyperbola.

Note: the circle with red dots surrounding G was used to construct the inscribed parabolae generated from lines through G.

Download a GSP document with an interactive version of this picture.