X(1), X(2), X(3), X(241), X(3229), X(5000), X(5001)
A', B', C' : midpoints of ABC
Ia, Ib, Ic : excenters
Ka, Kb, Kc : centers of Apollonian circles
E1, E2 on the figure are X(5000), X(5001)
The tangents at A, B, C are the medians and those at the in/excenters pass through O.
The fourth point on AI is A1 = b + c : b : c and the fourth point on IbIc is A2 = b - c : b : -c. These two points lie on the sideline B'C' of the medial triangle and the points B1, B2, C1, C2 are defined similarly.
Locus properties :
Q026 is the locus of point P such that
Let PaPbPc be the circumcevian triangle of P and let Ka, Kb, Kc be the centroids of PaBC, PbCA, PcAB respectively. PaPbPc and KaKbKc are perspective if and only if P lies on Q026.
Q026 is the quartic Q(X2) mentioned in page K1065 where other examples are given.
Q026 is a member of a family of quartics detailed at the bottom of page Q023.