Home page  Catalogue  Classes  Tables  Glossary  Notations  Links  Bibliography  Thanks  Downloads  Related Curves 

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

X(1), X(3), X(4), X(98), X(147), X(182), X(262), X(511) pivot : X(5999) = P422 = X(2)X(3) /\ X(98)X(511) infinite points of the circumellipse passing through X(99) and X(685) 

K422 is a an isogonal pK. Its pivot is P422, the harmonic conjugate of X(384) with respect to O and H. P422 is now X(5999), the common point of the lines X(2)X(3), X(98)X(385), X(147)X(325), X(182)X(262), etc. Thus, K422 is a member of the Euler pencil. See also Table 32, Kiepert AntiCevian mates. K422 is the unique circumcubic passing through the centers of the six Neuberg circles (points Ae, Be, Ce, Ai, Bi, Ci on the figures). These points are the vertices of the two Neuberg triangles, see Mathworld. The base angles of the six isosceles triangles are +/ (π/2  w), where w is the Brocard angle. 

Note that the lines AH, BeCi and BiCe concur on the cubic. Two other triads of lines similarly. The 1st (resp. 2nd) Neuberg triangle and the anticevian triangle of M are perspective if and only if M lies on K128 (resp. K423). In both cases, the locus of the perspector is K422. *** See another related property here (in Spanish, 20171119, Angel Montesdeoca). 
