HYACINTHOS 26876

[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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gX1312:

 

a^2 (a^4-2 a^2 b^2+b^4+a^2 c^2+b^2 c^2-2 c^4+(a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) J) (a^4+a^2 b^2-2 b^4-2 a^2 c^2+b^2 c^2+c^4+(a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) J):: 

on lines {{4,13414},{69,13415},{110,1114},{186,249},{248,15166},{265,1313},{2574,5622},{13434,14374}}.
on quartic Q120.
on cubic K934.
on Jerabek hyperbola.
isogonal conjugate of X(1312).
isogonal of the complement X(1114).
X(i)-cross conjugate of X(j) for these (i,j): {{3, 1114}, {6, 8116}, {2574, 110}}.
X(i)-isoconjugate of X(j) for these (i,j): {{1, 1312}, {92, 15167}, {2575, 2589}, {2579, 2593}, {2583, 8106}}.
cevapoint of X(1114) and X(14709).
barycentric product X(i)X(j) for these {i,j}: {{249, 1313}, {1114, 8116}, {1823, 2581}}.
barycentric quotient X(i)/X(j) for these {i,j}: {{6, 1312}, {184, 15167}, {1114, 2593}, {1313, 338}, {1823, 2583}, {2577, 2589}, {3284, 14500}, {15166, 125}}.

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gX1313:

 

a^2 (a^4-2 a^2 b^2+b^4+a^2 c^2+b^2 c^2-2 c^4-(a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) J) (a^4+a^2 b^2-2 b^4-2 a^2 c^2+b^2 c^2+c^4-(a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) J):: 

on lines {{4,13415},{69,13414},{110,1113},{186,249},{248,15167},{265,1312},{2575,5622},{13434,14375}}.
on quartic Q120.
on cubic K934.
on Jerabek hyperbola.
isogonal conjugate of X(1313).
isogonal of the complement X(1113).
X(i)-cross conjugate of X(j) for these (i,j): {{3, 1113}, {6, 8115}, {2575, 110}}.
X(i)-isoconjugate of X(j) for these (i,j): {{1, 1313}, {92, 15166}, {2574, 2588}, {2578, 2592}, {2582, 8105}}.
cevapoint of X(1113) and X(14710).
barycentric product X(i)X(j) for these {i,j}: {{249, 1312}, {1113, 8115}, {1822, 2580}}.
barycentric quotient X(i)/X(j) for these {i,j}: {{6, 1313}, {184, 15166}, {1113, 2592}, {1312, 338}, {1822, 2582}, {2576, 2588}, {3284, 14499}, {15167, 125}}.

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Best regards,
Peter Moses.