HYACINTHOS 27065

[Antreas P. Hatzipolakis]:

Let ABC be a triangle amd A'B'C' the pedal triangle of N.

 

Denote:

Na, Nb, Nc = the NPC centers of NBC, NCA, NAB, resp.

(Nab), (Nac) = the NPCs of NNaB', NNaC', resp.

(Nbc), (Nba) = the NPCs of NNbC', NNbA', resp.

(Nca), (Ncb) = the NPCs of NNcA', NNcB', resp.

 

R1 = the radical axis of (Nab), (Nac) 

R2 = the radical axis of (Nbc), (Nba)
R3 = the radical axis of (Nca), (Ncb)

A*B*C* = the triangle bounded by R1, R2, R3

ABC, A*B*C* are parallelogic.
The parallelogic center (ABC, A*B*C*) lies on the circuncircle.

 

 

[Peter Moses]:


Hi Antreas,

a^2 (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) (a^8 b^4-4 a^6 b^6+6 a^4 b^8-4 a^2 b^10+b^12+a^10 c^2-2 a^8 b^2 c^2+3 a^6 b^4 c^2-8 a^4 b^6 c^2+10 a^2 b^8 c^2-4 b^10 c^2-4 a^8 c^4+2 a^6 b^2 c^4+a^4 b^4 c^4-8 a^2 b^6 c^4+6 b^8 c^4+6 a^6 c^6+2 a^4 b^2 c^6+3 a^2 b^4 c^6-4 b^6 c^6-4 a^4 c^8-2 a^2 b^2 c^8+b^4 c^8+a^2 c^10) (a^10 b^2-4 a^8 b^4+6 a^6 b^6-4 a^4 b^8+a^2 b^10-2 a^8 b^2 c^2+2 a^6 b^4 c^2+2 a^4 b^6 c^2-2 a^2 b^8 c^2+a^8 c^4+3 a^6 b^2 c^4+a^4 b^4 c^4+3 a^2 b^6 c^4+b^8 c^4-4 a^6 c^6-8 a^4 b^2 c^6-8 a^2 b^4 c^6-4 b^6 c^6+6 a^4 c^8+10 a^2 b^2 c^8+6 b^4 c^8-4 a^2 c^10-4 b^2 c^10+c^12)::
on the circumcircle and the lines {{54,1291},{110,1157},{143,476},{925,13150},{930,1154},{1141,1510},...}.

Best regards,
Peter Moses.