HYACINTHOS 26875

[Antreas P. Hatzipolakis]:

 

In  K932 we read that the cubic passes through the points:

 

X(3), X(4), X(110), X(523), X(7471), X(14264), X(7471)*, X(14264)*

The two last ones X(7471)*, X(14264)* are isogonal conjugates of X(7471), X(14264), resp.

In K934 we read that the cubic passes through the points

X(4), X(30), X(143), X(1147), X(1992), X(2574), X(2575), isogonal conjugates of X(1312), X(1313).

Which are these isogonal conjugates of X(7471), X(14264) and X(1312), X(1313) ? (properties)

 

 

[Peter Moses]:

 

Hi Antreas,

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g X(7471): 

a^2*(b^2 - c^2)*(a^8 - a^4*b^4 - 2*a^2*b^6 + 2*b^8 - 4*a^6*c^2 + 4*a^2*b^4*c^2 - 2*b^6*c^2 + 6*a^4*c^4 - b^4*c^4 - 4*a^2*c^6 + c^8)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - a^4*c^4 + 4*a^2*b^2*c^4 - b^4*c^4 - 2*a^2*c^6 - 2*b^2*c^6 + 2*c^8):: 

on lines {{3, 526}, {6, 686}, {68, 9033}, {69, 3268}, {74, 924}, {125, 15328}, {265, 523}, {520, 5504}, {690, 4846}, {895, 8675}, {1510, 11559}, {2605, 10091}, {3154, 14220}, {3566, 10293}}.
reflection of X(15328) in X(125).
on cubic K932.
on Jerabek hyperbola.
isogonal conjugate of X(7471).
antigonal image of X(15328).
isogonal of the anticomplement X(3154).
X(i)-isoconjugate of X(j) for these (i,j): {{1, 7471}, {662, 3018}}.
X(250)-vertex conjugate of X(15395).
trilinear pole of line {647, 2088}.
barycentric quotient X(i)/X(j) for these {i,j}: {{6, 7471}, {512, 3018}}.

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g X(14264): 

(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 2*a^4*c^2 + 2*a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 + b^2*c^4)*(a^6 - 2*a^4*b^2 + a^2*b^4 - a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - 2*b^2*c^4 + c^6):: 

on lines {{2, 5627}, {3, 523}, {4, 110}, {20, 477}, {32, 3163}, {315, 5641}, {376, 1138}, {1272, 6337}, {1511, 14254}, {1990, 2420}, {3260, 10564}, {3522, 14536}, {6662, 11250}, {10653, 11080}, {10654, 11085}, {14264, 14611}}.
on cubics K009,K210,K567,K617,K932.
isogonal conjugate of X(14264).
X(1300)-Ceva conjugate of X(30).
X(1495)-cross conjugate of X(14910).
X(i)-isoconjugate of X(j) for these (i,j): {{1, 14264}, {74, 1725}, {2159, 3580}, {2349, 3003}}.
cevapoint of X(i) and X(j) for these (i,j): {{30, 1511}, {1495, 3163}, {3258, 9033}}.
trilinear pole of line {1637, 3284}.
crossdifference of every pair of points on line {686, 3003}.
barycentric product X(i)X(j) for these {i,j}: {{30, 2986}, {687, 9033}, {1300, 11064}, {2407, 15328}, {3260, 14910}, {4240, 15421}, {12028, 14920}}.
barycentric quotient X(i)/X(j) for these {i,j}: {{6, 14264}, {30, 3580}, {1495, 3003}, {1990, 403}, {2173, 1725}, {2420, 15329}, {2986, 1494}, {3163, 113}, {3284, 13754}, {5504, 14919}, {9033, 6334}, {9409, 686}, {14910, 74}, {15328, 2394}}.
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Best regards,
Peter Moses.