HYACINTHOS 24667

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

Oib, Oib, Oic = the circumcenters of INbNc, INcNa, INaNb, resp.

The orthocenter of OiaOibOic lies on the Euler line of ABC.

 

[Peter Moses]:

Hi Antreas,

>The orthocenter of OiaOibOic lies on the Euler line of ABC.
Isn't it N?

On the IN line, G of OiaOibOic isn't too bad though:

4 a^4-4 a^3 b-5 a^2 b^2+4 a b^3+b^4-4 a^3 c+8 a^2 b c-4 a b^2 c-5 a^2 c^2-4 a b c^2-2 b^2 c^2+4 a c^3+c^4:: on lines {{1,5},{2,5844},{3,3622},{8, 3628},{30,5603},{140,1482},{ 145,1656},{165,3653},...}.

 

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5,1483),(1,5901,5),(1,7951, 1317),(1,9624,355),(355,5886, 7988),(1482,3616,140),(7988, 9624,5886).

 

Reflection of X(i) and X(j) for these {i,j}: {{5,5886},{5657,140},{5790, 547},{5886,5901},{8703,3576}}.

 

Midpoint of X(i) and X(j) for these {i,j}: {{1,5886},{2,10247},{381,7967} ,{1482,5657},{1699,3655},{ 3241,5790},{3576,3656},{5603, 10246}}.

 

2 X[1] + X[5], 5 X[5] - 2 X[355], 5 X[1] + X[355], 2 X[140] + X[1482], 4 X[1] - X[1483], 2 X[5] + X[1483], 4 X[355] + 5 X[1483], 4 X[1387] - X[1484], X[145] + 5 X[1656], 2 X[140] - 5 X[3616], X[1482] + 5 X[3616], X[3] - 7 X[3622], X[8] - 4 X[3628], X[165] - 3 X[3653], 3 X[355] - 5 X[5587], 3 X[5] - 2 X[5587], 3 X[1] + X[5587], 3 X[1483] + 4 X[5587], 5 X[3616] - X[5657], 3 X[5603] + X[5731], 11 X[355] - 5 X[5881], 11 X[5587] - 3 X[5881], 11 X[5] - 2 X[5881], 11 X[1] + X[5881], 11 X[1483] + 4 X[5881], X[5881] - 11 X[5886], X[355] - 5 X[5886], X[5587] - 3 X[5886], X[1483] + 4 X[5886], X[355] - 10 X[5901], X[5587] - 6 X[5901], X[5] - 4 X[5901], X[1] + 2 X[5901], X[1483] + 8 X[5901], 5 X[5587] - 9 X[7988], 5 X[5] - 6 X[7988], X[355] - 3 X[7988], 5 X[5886] - 3 X[7988], 10 X[5901] - 3 X[7988], 5 X[1] + 3 X[7988], 5 X[1483] + 12 X[7988], 17 X[5] - 14 X[7989], 17 X[5886] - 7 X[7989], 17 X[1] + 7 X[7989], 7 X[5587] - 15 X[8227], 7 X[5] - 10 X[8227], 7 X[5886] - 5 X[8227], 14 X[5901] - 5 X[8227], 7 X[1] + 5 X[8227], 5 X[7989] - 17 X[9624], 5 X[5] - 14 X[9624], X[355] - 7 X[9624], 5 X[5886] - 7 X[9624], 10 X[5901] - 7 X[9624], 3 X[7988] - 7 X[9624], 5 X[1] + 7 X[9624], 5 X[5603] - X[9812], 5 X[5731] + 3 X[9812], X[5731] - 3 X[10246], X[9812] + 5 X[10246],...

 

Best regards,

Peter Moses.