Excerpt from On the in-and-Circumscribed Triangles of the Plane Rational Quartic Curve
The last and most difficult case is when the six curves are all of them one and the same carve.
It is to be noted that this formula gives the same number of triangles as has been found by the method used later. For example, in the case of the rational quartic, where a=4, A=6, a=18, the number of triangles is 8, which corresponds to that found on page 18. For the cuspidal quartic, where a=4, A=5, a=16, the number is two, which also corresponds to the number found on page 22.
In this paper it is proposed to look into the existence and actual number of such triangles for the following types of rational quartics:
I. Quartic with three double points.
II. Quartic with one double point and a tacnode.
III. Quartic with a triple point.
IV. Quartic with two double points and a cusp.
This discussion was led up to by preliminary work on the three-cusped rational quintic. Upon subjection to a quadratic transformation this curve goes into a rational quartic, which, it will be shown, has triangles of the kind here mentioned. Accordingly, it will first be proved that the quintic can have certain conditions imposed upon its coefficients so that it may acquire an additional cusp or a tacnode without degenerating. It will also be shown that it cannot have a triple point.
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