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X(1), X(4), X(84)
Ia, Ib, Ic vertices of excentral triangle
feet of altitudes
points at infinity of the altitudes
vertices of the circumnormal triangle
foci of the inconic with center O, perspector X(69)
Denote by PaPbPc the pedal triangle of point P and by Qa, Qb, Qc the intersections of the lines AP and PbPc, BP and PcPa, CP and PaPb respectively.
The triangles PaPbPc and QaQbQc are perspective if and only if P lies on the Darboux cubic (together with the line at infinity and the circumcircle).
This question was raised by Nikolaos Dergiades and answered by Antreas Hatzipolakis (Hyacinthos #8336 & sq.) who asks the locus of P such that QaQbQc and ABC are orthologic.
The sought locus is Q009, a bicircular circum-septic whose isogonal conjugate is Q010.
It has three real asymptotes which are the altitudes.
The tangents at the in/excenters pass through H.