HYACINTHOS 26442

[Tran Quang Hung]:(rephrased)

 

Let ABC be a triangle and A'B'C' the antipedal triangle of N.

Denote:

Na, Nb, Nc = the NPC centers of the triangles NB'C', NC'A', NA'B', resp.

Ha, Hb, Hc =the orthocenters of the triangles NB'C',NC'A',NA'B', resp.

Naa, Nbb, Ncc = the NPC centers of the triangles NHbHc,NHcHa,NHaHb, resp.

1. The NPC center of the triangle NaNbNc lies on Euler line of ABC.
Which is this point ?

2. (NaNbNc, NaaNbbNcc) are orthologic.

Orthologic centers?

[Peter Moses]:

Hi Antreas,

1). X(5500)

2).
(NaNbNc, NaaNbbNcc) orthology: X(10205).

(NaaNbbNcc, NaNbNc) orthology:
2 a^22-11 a^20 b^2+20 a^18 b^4-a^16 b^6-48 a^14 b^8+70 a^12 b^10-28 a^10 b^12-30 a^8 b^14+46 a^6 b^16-27 a^4 b^18+8 a^2 b^20-b^22-11 a^20 c^2+50 a^18 b^2 c^2-98 a^16 b^4 c^2+137 a^14 b^6 c^2-185 a^12 b^8 c^2+179 a^10 b^10 c^2-45 a^8 b^12 c^2-97 a^6 b^14 c^2+108 a^4 b^16 c^2-45 a^2 b^18 c^2+7 b^20 c^2+20 a^18 c^4-98 a^16 b^2 c^4+212 a^14 b^4 c^4-246 a^12 b^6 c^4+104 a^10 b^8 c^4+47 a^8 b^10 c^4+34 a^6 b^12 c^4-162 a^4 b^14 c^4+110 a^2 b^16 c^4-21 b^18 c^4-a^16 c^6+137 a^14 b^2 c^6-246 a^12 b^4 c^6+96 a^10 b^6 c^6+19 a^8 b^8 c^6+12 a^6 b^10 c^6+108 a^4 b^12 c^6-160 a^2 b^14 c^6+35 b^16 c^6-48 a^14 c^8-185 a^12 b^2 c^8+104 a^10 b^4 c^8+19 a^8 b^6 c^8+10 a^6 b^8 c^8-27 a^4 b^10 c^8+170 a^2 b^12 c^8-34 b^14 c^8+70 a^12 c^10+179 a^10 b^2 c^10+47 a^8 b^4 c^10+12 a^6 b^6 c^10-27 a^4 b^8 c^10-166 a^2 b^10 c^10+14 b^12 c^10-28 a^10 c^12-45 a^8 b^2 c^12+34 a^6 b^4 c^12+108 a^4 b^6 c^12+170 a^2 b^8 c^12+14 b^10 c^12-30 a^8 c^14-97 a^6 b^2 c^14-162 a^4 b^4 c^14-160 a^2 b^6 c^14-34 b^8 c^14+46 a^6 c^16+108 a^4 b^2 c^16+110 a^2 b^4 c^16+35 b^6 c^16-27 a^4 c^18-45 a^2 b^2 c^18-21 b^4 c^18+8 a^2 c^20+7 b^2 c^20-c^22::

Best regards,
Peter Moses.