Abstract
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.
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van de Craats, J. An unexpected appearance of Steiner's hypocycloid. Aeq. Math. 30, 239–251 (1986). https://doi.org/10.1007/BF02189930
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DOI: https://doi.org/10.1007/BF02189930