HYACINTHOS 26822

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the cevian triangle of O.

The NPC centers Na, Nb, Nc of NAA', NBB', NCC', resp. are collinear.

The line NaNbNc is perpendicular to Euler line.

Point of intersection?

 

 

[Peter Moses]:

 

 

Hi Antreas,

 

2 a^10-7 a^8 b^2+6 a^6 b^4+4 a^4 b^6-8 a^2 b^8+3 b^10-7 a^8 c^2+6 a^6 b^2 c^2-7 a^4 b^4 c^2+17 a^2 b^6 c^2-9 b^8 c^2+6 a^6 c^4-7 a^4 b^2 c^4-18 a^2 b^4 c^4+6 b^6 c^4+4 a^4 c^6+17 a^2 b^2 c^6+6 b^4 c^6-8 a^2 c^8-9 b^2 c^8+3 c^10:: 

on lines {{2,3},{1154,12900},{5447,1344 6}}.

Midpoint of X(i) and X(j) for these {i,j}: {{140, 403}, {2072, 10096}, {5447, 13446}}.

3 X[5] + X[186], 5 X[632] - X[2071], 3 X[547] - X[2072], X[2070] + 7 X[3090], X[3153] - 9 X[5055], 5 X[3628] - 2 X[5159], 15 X[1656] + X[5899], 3 X[5189] + 5 X[5899], 5 X[547] + X[7426], 5 X[2072] + 3 X[7426], 3 X[7426] - 5 X[10096], 3 X[547] + X[10096], 13 X[5] - X[10296], 13 X[186] + 3 X[10296], 3 X[403] - X[11558], 3 X[140] + X[11558], 3 X[2] + X[11563], X[7575] + 5 X[12812], 7 X[3851] + X[13619].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 14940, 140), (140, 546, 3520), (547, 10096, 2072).


Best regards,
Peter Moses.