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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

1. the anticevian and pedal triangles of P are orthologic.
2. the anticevian and reflection triangles of P are orthologic.
3. the circumcevian triangle of P and the anticevian triangle of the isogonal conjugate of P are perspective.
4. the circumcevian triangle of P and the circumanticevian triangle of the isogonal conjugate of P are perspective.
5. the circumcevian triangle of P and the medial triangle are perspective, together with the circumcircle (Angel Montesdeoca, 2022-10-26). The locus of the perspector is a complicated 10th degree curve.

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.