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A, B, C (double)
Ia, Ib, Ic : excenters
vertices of the cevian triangle of K
4 foci of the Steiner in-ellipse (two real F1, F2)
4 intersections of the Stammler hyperbola and the circumcircle (two real S1, S2)
2 intersections T1, T2 of the line GK with the Steiner ellipse
Let P be a point and let G', G" be the centroids of the pedal and antipedal triangles of P respectively. P, G', G" are collinear if and only if P lies on the quintic Q017 or on the line at infinity.
Q017 has three singularities at A, B, C and contains the in/excenters.
The "fifth" points on the bisectors at A are A5 = a^2 : 2b(b+c) : 2c(b+c) on AI and A5' = a^2 : 2b(b-c) : -2c(b-c) on IbIc. These points are collinear with the midpoint of BC.
The "fifth" point on the symmedian AK is -2a^2 : b^2 : c^2.
K is a flex with inflexional tangent GK, the tangent at K to the Stammler hyperbola.