HYACINTHOS 25750

 

[Tran Quang Hung]:

Let ABC be a triangle with NPC center N.

 

Na,Nb,Nc are NPC centers of triangle NBC, NCA, NAB, resp.

 

NbNc cuts NA at A'.

 

Similar we have B',C'.

 

The orthocenter of triangle A'B'C' lies on Euler line of ABC.

 

Which is this point ?

 

 

[Peter Moses]:

Hi Antreas,

 

2 a^16-5 a^14 b^2+a^12 b^4+5 a^10 b^6-5 a^8 b^8+13 a^6 b^10-25 a^4 b^12+19 a^2 b^14-5 b^16-5 a^14 c^2+10 a^12 b^2 c^2-3 a^10 b^4 c^2-10 a^8 b^6 c^2-3 a^6 b^8 c^2+54 a^4 b^10 c^2-69 a^2 b^12 c^2+26 b^14 c^2+a^12 c^4-3 a^10 b^2 c^4+16 a^8 b^4 c^4-7 a^6 b^6 c^4-44 a^4 b^8 c^4+93 a^2 b^10 c^4-56 b^12 c^4+5 a^10 c^6-10 a^8 b^2 c^6-7 a^6 b^4 c^6+30 a^4 b^6 c^6-43 a^2 b^8 c^6+70 b^10 c^6-5 a^8 c^8-3 a^6 b^2 c^8-44 a^4 b^4 c^8-43 a^2 b^6 c^8-70 b^8 c^8+13 a^6 c^10+54 a^4 b^2 c^10+93 a^2 b^4 c^10+70 b^6 c^10-25 a^4 c^12-69 a^2 b^2 c^12-56 b^4 c^12+19 a^2 c^14+26 b^2 c^14-5 c^16::

on lines {{2, 3}, {137, 11801}}.

 

Best regards,

Peter Μoses.