Dear all,

If A'B'C' is the pedal triangle of P (instead of cevian triangle of P), and A", B", C" as before,

If A* is the homothetic of A" under h(A', k), B* and C* likewise, then the locus of P such that ABC and A*B*C* are perspective (at X) is a central cubic passing through X3 with asymptotes parallel to the altitudes of ABC and concurring on the Brocard axis, very similar to the Darboux cubic which is obtained when k = 0.

The locus of the perspector X is pK(X2, T) where

T = (-a^4+b^4+c^4)k-4 SA Sw : : , on the line X4-X69.

If k = 0, T = X69 and this is the Lucas cubic.

Best regards

Bernard

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