Dear Antreas, Jean-Pierre, Fred and friends,

Let P be a point with traces X,Y,Z on the sides of triangle ABC.

B_(X), C_(X) the perpendicular feet of X on CA and AB respectively.

Similarly define C_(Y), A_(Y), A_(Z), and B_(Z). The locus of

P for which the triangle bounded by the three lines B_(X)C_(X),

C_(Y)A_(Y), and A_(Z)B_(Z) is perspective is the union of the

following three curves.

(1) The circumcircle, In this case, the three lines

are parallel to the Simson line of P.

(2) The Darboux cubic.

(3) The quartic with equation

sum aaSBC v^2w^2 = (sum aaSBC)(u+v+w)uvw.

In this case, the three lines are concurrent.

The quartic in (3) can be constructed as follows. It is the

ISOTOMIC conjugate of a conic C, which is, with respect to the

anticomplementary triangle, the circumconic with perspector

the pro-orthocenter, i.e., the point with homogeneous barycentric

coordinates

(a^2/SA : b^2/SB : c^2/SC).

Best regards.

Sincerely,

Paul