HYACINTHOS 26810

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point

Denote:

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

NaaNabNac  = the midway triangle of Na
(ie Naa, Mab, Nac = the midpoints of ANa, BNa, CNa, resp.)
NbaNbbNbc  = the midway triangle of Nb

NcaNcbNcc  = the midway triangle of Nc

Oa, Ob, Oc = the circumcenters of NaaNabNac, NbaNbbNbc, NcaNcbNcc, resp.

1. For P = I
The orthocenter of OaObOc is the midpoint of OI

2. For P = N
The circumcenter of OaObOc lies on the Euler line of ABC.

 

3. For P = O:
The NPC center of OaObOc lies on the Euler line of ABC.

 

 

 

[Angel Montesdeoca]:

***  1. For P = I
    The orthocenter of OaObOc is the midpoint of OI:   X(1385)

***    2. For P = N
    The circumcenter of OaObOc lies on the Euler line of ABC:   midpoint of X(3) and X(5501) =  9th Hatzipolakis-Montesdeoca Point
   
-(b^2-c^2)^6 (b^4-4 b^2 c^2+c^4)+7 (b^2-c^2)^4 (b^6-3 b^4 c^2-3 b^2 c^4+c^6) a^2-(b^2-c^2)^2 (29 b^8-36 b^6 c^2-41 b^4 c^4-36 b^2 c^6+29 c^8) a^4+(73 b^10-55 b^8 c^2-27 b^6 c^4-27 b^4 c^6-55 b^2 c^8+73 c^10) a^6+(-105 b^8-58 b^6 c^2-40 b^4 c^4-58 b^2 c^6-105 c^8) a^8+(81 b^6+89 b^4 c^2+89 b^2 c^4+81 c^6) a^10-3 (9 b^4+10 b^2 c^2+9 c^4) a^12+(-b^2-c^2) a^14+2 a^16: ... : ....

On line {2.3}
midpoint of  X(3) and X(5501)
   
     (6 - 9 - 13) - search numbers (1.79733458884568, 0.923073251920569, 2.17207472803367)
    
    
    
***    3. For P = O:
    The NPC center of OaObOc lies on the Euler line of ABC: X(10212)=  35th  Hatzipolakis-Montesdeoca Point

Angel Montedeoca