HYACINTHOS 25023

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

Na, Nb, Nc= the NPC centers of NBC, NCA, NAB, resp.

N1, N2, N3 = the orthogonal projections of Na, Nb, Nc on OA, OB, OC, resp.

ABC, N1N2N3 are orthologic.

Generalization:

Let A*, B*, C* be points on NaN1, NbN2, NcN3, resp. such that:

A*N1/A*Na = B*N2/B*Nb = C*N3/C*Nc = t

ABC, A*B*C* are orthologic.

[Peter Moses]:

Hi Antreas,

 

>ABC, N1N2N3 are orthologic.

 

(2 a^8-3 a^6 b^2+2 a^4 b^4-3 a^2 b^6+2 b^8-4 a^6 c^2+3 a^4 b^2 c^2+3 a^2 b^4 c^2-4 b^6 c^2-4 a^2 b^2 c^4+4 a^2 c^6+4 b^2 c^6-2 c^8) (2 a^8-4 a^6 b^2+4 a^2 b^6-2 b^8-3 a^6 c^2+3 a^4 b^2 c^2-4 a^2 b^4 c^2+4 b^6 c^2+2 a^4 c^4+3 a^2 b^2 c^4-3 a^2 c^6-4 b^2 c^6+2 c^8)::
on lines {{3,9140},{6,7699},{125,3431}, {3426,10721},{3448,4846}}.

isogonal of X(7575).

reflection of X(3431) in X(125).

3 X[7699] - 2 X[10294].

on Jerabek.

vertex conjugate of X(54) and X(1177).

cevapoint of X(125) and X(9003)

trilinear pole of line X(566) X(647).

 

and the other

 

4 a^10-9 a^8 b^2+5 a^6 b^4-a^4 b^6+3 a^2 b^8-2 b^10-9 a^8 c^2+6 a^6 b^2 c^2+4 a^4 b^4 c^2-7 a^2 b^6 c^2+6 b^8 c^2+5 a^6 c^4+4 a^4 b^2 c^4+8 a^2 b^4 c^4-4 b^6 c^4-a^4 c^6-7 a^2 b^2 c^6-4 b^4 c^6+3 a^2 c^8+6 b^2 c^8-2 c^10::
on lines {{3,9140},{1154,6146},{1493, 7574},{6240,6746},{6756,10095} }.

2 X[6756] - 3 X[10095], 3 X[6146] - X[11264].

 

Best regards,

Peter Moses.