An unexpected appearance of Steiner's hypocycloid

J. de Cicco 1939] observed that two parabolas must touch each other if they have parallel axes, while one parabola touches the three sides of a given triangle and the other passes through the midpoints of those sides. Coxeter 1983] showed that the locus of the point of contact of the two parabolas, if the triangle is kept fixed while the common axial direction varies, is a rational cubic curve. In a subsequent paper, Coxeter investigated other aspects of this cubic (Coxeter 1985]).De Cicco's theorem, viewed as a result in the projective plane, can be dualized in a natural way. The cubic then becomes a set of lines, enveloping a curve of class three. We shall show that this curve is a quartic curve with three cusps, which is projectively equivalent to Steiner's well-known hypocycloid.