HYACINTHOS 26170

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

N1, N2, N3 = the NPC centers of OBC, OCA, OAB, resp.

Na, Nb, Nc = the reflections of N1, N2, N3 in AO, BO, CO, resp.

The circumcenter of N1N2N3 lies on the Euler line.

[Peter Moses]:

 

Hi Antreas,

 

>The circumcenter of N1N2N3 lies on the Euler line.

X(10226) is the circumcenter of NaNbNc lies on the Euler line.

 

>The circumcenter of N1N2N3, 

a^6 b^4-3 a^4 b^6+3 a^2 b^8-b^10-2 a^6 b^2 c^2+2 a^4 b^4 c^2-3 a^2 b^6 c^2+3 b^8 c^2+a^6 c^4+2 a^4 b^2 c^4-2 b^6 c^4-3 a^4 c^6-3 a^2 b^2 c^6-2 b^4 c^6+3 a^2 c^8+3 b^2 c^8-c^10::, 

lies on lines {{2,156},{3,12278},{5, 113},{26,1853},{30,5449},...}.

Complement X[156].

midpoint of X(i) and X(j) for these {i,j}: {{9927,11250},{12359,13371}}.

reflection of X(i) in X(j) for these {i,j}: {{10282,10125},{12038, 5498}}.

X[26] + 3 X[1853].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,10264,185),(3448,6143,49).

Searches: {4. 53932730242720195010896651206, 4. 29323618897705372197243439169, -1. 42665009619690514291285248193} .

 

Best regards,

Peter Moses.