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too complicated to be written here. Click on the link to download a text file. 

X(3), X(4) A', B', C' : midpoints Ha, Hb, Hc : feet of the altitudes A1, B1, C1 : projections of O on the altitudes E1, E2 see below 

Denote by A1, B1, C1 the reflections of P in A, B, C and by A3, B3, C3 the reflections of H in the lines AP, BP, CP. The three circles AB1C1, BC1A1, CA1B1 have a common point D on the circumcircle. The three circles PAA3, PBB3, PCC3 have P in common and another point L also on the circumcircle. These points D, L are antipodes if and only if P lies on the Thomson cubic (Musselman, Some loci connected with a triangle. Monthly, p.354361, JuneJuly 1940). On the other hand, they coincide if and only if P lies on the bicicular quartic Q013 we call the Musselman quartic. Q013 is also the locus of M whose orthotransversal and polar in the circumcircle are perpendicular. When these are parallel, we find the Orthocubic K006. Q013 is a bicircular quartic which is tritangent at A, B, C to the circumcircle (O). It is also bitangent to the line at infinity at the circular points. Q013 has the following properties :

