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X(1), X(2), X(4), X(254), X(1113), X(1114), X(1138), X(2184), X(3223), X(3346), X(3459) excenters, vertices of the antimedial triangle more generally, vertices of the anticevian triangle of any of its points 

Q066 is the isogonal transform of the Stammler rectangular hyperbola. This is the diagonal hyperbola passing through X(1), X(3), X(6), X(155), X(159), X(195), X(399), X(610), X(1498), X(1740), X(2574), X(2575), X(2916), X(2917), X(2918), X(2929), X(2930), X(2931), X(2935), X(2948) and the vertices of the anticevian triangle of any of its points. Q066 is also the square root of the circumconic passing through X(2), X(6), X(25), X(37), X(42), X(111), X(251), X(263), X(308), X(393), X(493), X(494), X(588), X(589), X(694), X(941), X(967), X(1169), X(1171), X(1218), X(1239), X(1241), X(1383), X(1400), X(1427), X(1880), X(1976), X(1989), X(2054), X(2165), X(2248), X(2350), X(2395), X(2433), X(2963), X(2981), X(2987), X(2998). In other words, Q066 contains the square roots of all the points on the conic. Q066 is also the cyclocevian transform of the Kiepert hyperbola i.e. the cyclocevian conjugate M' of a point M on the Kiepert hyperbola ends up on Q066. Here is a list of such pairs {M,M'} : {2,4}, {4,2}, {13,13483}, {14,13484}, {1029,1}, {6504,3346}, {6625,8049}, {13576,9510}. (Peter Moses, 20170615). See property 4 below. The points A, B, C are nodes. The nodal tangents pass through the traces of the circles orthogonal to the circumcircle and centered at the intersections of the perpendiculars at A, B, C to the Brocard axis with the sidelines of ABC. Q066 is the locus of point P such that
Other locus properties :




X(1138) is the isogonal conjugate of X(399), the Parry reflection point. These two points lie on the Neuberg cubic. JeanPierre Ehrmann has shown that, except the trivial case H, X(1138) is the only point such that the (blue) pedal and (orange) cevian triangles are (indirectly) similar. These triangles are also simultaneously orthologic and parallelogic. See Table 7 and Table 8. 

Orthologic pedal and cevian triangles of X(1138) with centers of orthology O1 and O2 
Parallelogic pedal and cevian triangles of X(1138) with centers of parallelogy P1 and P2 

*** The same facts remain true with the cevian and reflection triangles of X(1138) : these two triangles are simultaneously (indirectly) similar, orthologic and parallelogic. One of the centers of orthology (resp. parallelogy) is O1 (resp. P1). 



(contributed by Angel Montesdeoca, 20160127) Let F be a finite point. The locus of P whose cevian circle passes through F is generally a sextic S(F) and the isogonal transform of S(F) is a cubic K(F). The discriminant of K(F) vanishes (i.e. K(F) has at least one double point) if and only if : • F lies on one in/excircle in which case K(F) is in general a nodal cubic with node X(55) or extraversion. • F lies on the nine point circle in which case K(F) splits into a line passing through X(3) and a circumconic passing through X(6). Hence S(F) is the union of a rectangular circumhyperbola H(F) with center F and a circumquartic Q(F) passing through X(2) with nodes at A, B, C. When F = X(115), Q(F) is Q066 and H(F) is the Kiepert hyperbola as said above. When F = X(125), Q(F) is Q124 and H(F) is the Jerabek hyperbola. Note that Q066 is the barycentric product of Q124 by X(1). See here for further examples. See also Q182, a similar quartic, the barycentric product of Q124 by X(6). 
