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X(1), X(2), X(3), X(241), X(3229), X(5000), X(5001)

midpoints of ABC

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 b + c : b : c and the fourth point on IbIc is b - c : b : -c.

Locus properties :

Q026 is the locus of point P such that

  • the anticevian and pedal triangles of P are orthologic.
  • the anticevian and reflection triangles of P are orthologic.
  • the circumcevian triangle of P and the anticevian triangle of the isogonal conjugate of P are perspective.
  • the circumcevian triangle of P and the circumanticevian triangle of the isogonal conjugate of P are perspective.

Q026 is the quartic Q(X2) mentioned in page K1065 where other examples are given.