Peter Yff, A family of quartics associated with a triangle,
Forum Geometricorum, 9 (2009) 165--171.
Abstract: It is known that the envelope of the family of pedal lines
(Simson or Wallace lines) of a triangle ABC is Steiner's deltoid,
a three-cusped hypocycloid that is concentric with the nine-point circle
of ABC and touches it at three points. Also known is that the nine-point
circle is the locus of the intersection point of two perpendicular
pedal lines. This paper considers a generalization in which two pedal
lines form any acute angle \theta. It is found that the locus of their
intersection point, for any value of \theta, is a quartic curve with
the same axes of symmetry as the deltoid. Moreover, the deltoid is the envelope
of the family of quartics. Finally, it is shown that all of these quartics,
as well as the deltoid and the nine-point circle, may be simultaneously
generated by points on a circular disk rolling on the inside of a fixed
circle.
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