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Re: [EMHL] Re: Happy New Year and Nik's locus for 2013

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  • Bernard Gibert
    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*
    Message 1 of 14 , Jan 2, 2013
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      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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