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6 feet of bisectors

Ia, Ib, Ic vertices of excentral triangle

vertices of the circumnormal triangle

foci of the inconic with center O, perspector X(69) : F1, F2 and two imaginary

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 is a pedal triangle with respect to PaPbPc.

The sought locus is Q007, a circular circum-octic whose isogonal conjugate is Q008.

The vertices of ABC are triple points. The tangents at A to Q007 are the cevian line of X(25) and two other perpendicular meeting BC at U1, U2. These points lie on the circle centered at Ao (cevian of O) passing through A.

The tangents at the in/excenters are those of the McCay cubic and therefore pass through O.

The 24 common points of Q007 and the McCay cubic are A, B, C (counting as 9 points), the vertices of the circumnormal triangle, the in/excenters (counting as 8 points), the four foci of the in-conic with center O. These latter points also lie on K019, the Brocard (third) cubic.