On the in-and-Circumscribed Triangles of the Plane Rational Quartic Curve #229523

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 trans formation 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.