## Re: locus related to Darboux cubic

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• A related problem: what is the locus of the centroid of the antipedal triangle of P, for P on the Darboux cubic? It would include X(2), X(154), X(165),
Message 1 of 5 , Mar 18, 2013
A related problem: what is the locus of the centroid of the antipedal triangle of P, for P on the Darboux cubic? It would include X(2), X(154), X(165), X(3158), and X(3167).

What points are in common with the first locus, besides X(2) and X(154)?

Randy

--- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:
>
> Hello, Randy,
>
> it is a cubic, but it doesn't go through the vertices.
>
> One of the three points where the cubic intersects the line BC is the point X1={0, 3 a^2 - b^2 + c^2, 3 a^2 + b^2 - c^2} that satisfies DX:XM=-4, where D is the foot of the A-altitude and M is the midpoint of BC.
>
> Equation of the cubic:
> -a^4 b^2 x^3 - 2 a^2 b^4 x^3 + 3 b^6 x^3 + a^4 c^2 x^3 -
> 13 b^4 c^2 x^3 + 2 a^2 c^4 x^3 + 13 b^2 c^4 x^3 - 3 c^6 x^3 -
> 3 a^6 x^2 y - 3 a^2 b^4 x^2 y + 6 b^6 x^2 y - 21 a^4 c^2 x^2 y +
> 9 b^4 c^2 x^2 y - 45 a^2 c^4 x^2 y - 48 b^2 c^4 x^2 y -
> 3 c^6 x^2 y - 6 a^6 x y^2 + 3 a^4 b^2 x y^2 + 3 b^6 x y^2 -
> 9 a^4 c^2 x y^2 + 21 b^4 c^2 x y^2 + 48 a^2 c^4 x y^2 +
> 45 b^2 c^4 x y^2 + 3 c^6 x y^2 - 3 a^6 y^3 + 2 a^4 b^2 y^3 +
> a^2 b^4 y^3 + 13 a^4 c^2 y^3 - b^4 c^2 y^3 - 13 a^2 c^4 y^3 -
> 2 b^2 c^4 y^3 + 3 c^6 y^3 + 3 a^6 x^2 z + 21 a^4 b^2 x^2 z +
> 45 a^2 b^4 x^2 z + 3 b^6 x^2 z + 48 b^4 c^2 x^2 z +
> 3 a^2 c^4 x^2 z - 9 b^2 c^4 x^2 z - 6 c^6 x^2 z - 24 a^4 b^2 x y z +
> 24 a^2 b^4 x y z + 24 a^4 c^2 x y z - 24 b^4 c^2 x y z -
> 24 a^2 c^4 x y z + 24 b^2 c^4 x y z - 3 a^6 y^2 z -
> 45 a^4 b^2 y^2 z - 21 a^2 b^4 y^2 z - 3 b^6 y^2 z -
> 48 a^4 c^2 y^2 z + 9 a^2 c^4 y^2 z - 3 b^2 c^4 y^2 z + 6 c^6 y^2 z +
> 6 a^6 x z^2 + 9 a^4 b^2 x z^2 - 48 a^2 b^4 x z^2 - 3 b^6 x z^2 -
> 3 a^4 c^2 x z^2 - 45 b^4 c^2 x z^2 - 21 b^2 c^4 x z^2 -
> 3 c^6 x z^2 + 3 a^6 y z^2 + 48 a^4 b^2 y z^2 - 9 a^2 b^4 y z^2 -
> 6 b^6 y z^2 + 45 a^4 c^2 y z^2 + 3 b^4 c^2 y z^2 +
> 21 a^2 c^4 y z^2 + 3 c^6 y z^2 + 3 a^6 z^3 - 13 a^4 b^2 z^3 +
> 13 a^2 b^4 z^3 - 3 b^6 z^3 - 2 a^4 c^2 z^3 + 2 b^4 c^2 z^3 -
> a^2 c^4 z^3 + b^2 c^4 z^3 = 0.
>
> --- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@> wrote:
> >
> > Friends,
> >
> > Let P be a point on the Darboux cubic. What is the locus of the centroid of the pedal triangle of P? It would include x(2), X(51), X(154), X(210), X(354), X(1853), and X(3917).
> >
> > Randy
> >
>
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