A Family of Quartics Associated with a Triangle

It is known [1, p.115] 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 [2, p.249] 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 θ. It is found that the locus of their intersection point, for any value of θ, 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. 1. Sketching the loci Consider two pedal lines of triangle ABC which intersect and form an angle θ. It is required to find the locus of the intersection point for all such pairs of pedal lines for any fixed value of θ. There are infinitely many loci as θ varies between 0 and π 2 . By plotting points, some of the loci are sketched in Figure 1. These include the cases θ = π 4 , π 3 , 5π 12 , and π 2 , the curves have been colored. As θ → 0, the locus approaches Steiner’s deltoid. It will be shown later that in general the locus is a quartic curve. As θ → π 2 , the quartic merges into two coincident circles (the nine-point circle). Otherwise each curve has three double points, which seem to merge into a triple point when θ = π 3 . This case resembles the familiar trefoil, or “three-leaved rose” of polar coordinates. 2. A conjecture Figure 1 seems to suggest that all of the loci might be generated simultaneously by points on a circular disk that rolls inside a fixed circle concentric with the nine-point circle. For example, the deltoid could be generated by a point on the circumference of the disk, provided that the radius of the disk is one third that of the circle. The other curves might be hypotrochoids generated by interior points of the disk. However, this fails because, for example, there is no generating point for the nine-point circle. Publication Date: July 27, 2009. Communicating Editor: Paul Yiu. 166 P. Yff Another possible approach is given by Zwikker [2, pp.248–249], who shows that the same hypocycloid of three cusps may be generated when the radius of the rolling circle is two thirds of the radius of the fixed circle. In this case the deltoid is generated in the opposite sense, and two circuits of the rolling circle are required to generate the entire curve. Simultaneously the nine-point circle is generated by the center of the rolling disk. It is now necessary to prove that every locus in the family is generated by a point on the rolling disk.