HYACINTHOS 26859

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle and P, P* be two isogonal conjugate points.

Denote:

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

N1, N2, N3 = the NPC centers of P*BC, P*CA, P*AB, resp.

N', N" = the NPC centers of NaNbNc, N1N2N3, resp.

Which is the locus of P such that N, N', N" are collinear?

Part of the locus is the Napoleon cubic.

***********

Let P = X(5) = N, P* = X(54)

Which points are the N', N" lying on the line X(5)X(54) ?

 

[Peter Moses]:

Hi Antreas,

>Which is the locus of P such that N, N', N" are collinear?

Napoleon + a degree 6.

>Let P = X(5) = N, P* = X(54)
>Which points are the N', N" lying on the line X(5)X(54) ?

N'=
2 a^16-7 a^14 b^2-a^12 b^4+41 a^10 b^6-85 a^8 b^8+83 a^6 b^10-43 a^4 b^12+11 a^2 b^14-b^16-7 a^14 c^2-2 a^12 b^2 c^2+53 a^10 b^4 c^2-50 a^8 b^6 c^2-55 a^6 b^8 c^2+112 a^4 b^10 c^2-63 a^2 b^12 c^2+12 b^14 c^2-a^12 c^4+53 a^10 b^2 c^4-48 a^8 b^4 c^4-19 a^6 b^6 c^4-56 a^4 b^8 c^4+123 a^2 b^10 c^4-52 b^12 c^4+41 a^10 c^6-50 a^8 b^2 c^6-19 a^6 b^4 c^6-26 a^4 b^6 c^6-71 a^2 b^8 c^6+116 b^10 c^6-85 a^8 c^8-55 a^6 b^2 c^8-56 a^4 b^4 c^8-71 a^2 b^6 c^8-150 b^8 c^8+83 a^6 c^10+112 a^4 b^2 c^10+123 a^2 b^4 c^10+116 b^6 c^10-43 a^4 c^12-63 a^2 b^2 c^12-52 b^4 c^12+11 a^2 c^14+12 b^2 c^14-c^16:: 

on lines {{5,49},{1656,14143},{3628,10615},{5501,13856}}.
midpoint of X(5501) and X(13856).
5 X[1656] - X[14143].


N''=
2 a^16-9 a^14 b^2+13 a^12 b^4-a^10 b^6-15 a^8 b^8+13 a^6 b^10-a^4 b^12-3 a^2 b^14+b^16-9 a^14 c^2+22 a^12 b^2 c^2-7 a^10 b^4 c^2-10 a^8 b^6 c^2-9 a^6 b^8 c^2+16 a^4 b^10 c^2+a^2 b^12 c^2-4 b^14 c^2+13 a^12 c^4-7 a^10 b^2 c^4-19 a^6 b^6 c^4-6 a^4 b^8 c^4+15 a^2 b^10 c^4+4 b^12 c^4-a^10 c^6-10 a^8 b^2 c^6-19 a^6 b^4 c^6-18 a^4 b^6 c^6-13 a^2 b^8 c^6+4 b^10 c^6-15 a^8 c^8-9 a^6 b^2 c^8-6 a^4 b^4 c^8-13 a^2 b^6 c^8-10 b^8 c^8+13 a^6 c^10+16 a^4 b^2 c^10+15 a^2 b^4 c^10+4 b^6 c^10-a^4 c^12+a^2 b^2 c^12+4 b^4 c^12-3 a^2 c^14-4 b^2 c^14+c^16:: 

on lines {{5,49},{6689,13365}}.

Best regards,
Peter Moses.