HYACINTHOS 25752

 [Antreas P. Hatzipolakis]:

Orthic version:

Let ABC be a triangle and A'B'C' the pedal triangle of H.

Denote:

Na, Nb, Nc = the NPC centers of HB'C', HC'A', HA'B', resp.

N1, N2, N3 = the NPC centers of AB'C', BC'A', CA'B', resp.

The circumcircles of HNaN1, HNbN2, HNcN3 are coaxial.

2nd point of intersection?

Excentral version Hyacinthos #/25216

 

[Peter Moses]:

Hi Antreas,

 

Orthic).
a^2 (a^12 b^2-4 a^10 b^4+5 a^8 b^6-5 a^4 b^10+4 a^2 b^12-b^14+a^12 c^2-4 a^10 b^2 c^2+7 a^8 b^4 c^2-11 a^6 b^6 c^2+16 a^4 b^8 c^2-13 a^2 b^10 c^2+4 b^12 c^2-4 a^10 c^4+7 a^8 b^2 c^4+2 a^6 b^4 c^4-8 a^4 b^6 c^4+9 a^2 b^8 c^4-6 b^10 c^4+5 a^8 c^6-11 a^6 b^2 c^6-8 a^4 b^4 c^6+3 b^8 c^6+16 a^4 b^2 c^8+9 a^2 b^4 c^8+3 b^6 c^8-5 a^4 c^10-13 a^2 b^2 c^10-6 b^4 c^10+4 a^2 c^12+4 b^2 c^12-c^14)::
on lines {{4,94},{30,11800},{52,10264}, {54,1511},{74,10263},{110, 5946},{113,10095},{125,1154},{ 182,12893},{381,12284},{389, 6153},{399,3567},{974,11565},{ 1199,11597},{1493,3043},...}.

Midpoint of X(i) and X(j) for these {i,j}: {{52, 10264}, {74, 10263}, {265, 6102}, {11800, 11806}}.

Reflection of X(i) in X(j) for these {i,j}: {{113,10095},{143,12236},{ 1511,12006},{10272,5462},{ 10627,6699},{11561,389}}.

3 X[143] - 2 X[1112], 3 X[568] + X[3448], X[399] - 5 X[3567], X[110] - 3 X[5946], 3 X[6102] - X[7722], 3 X[265] + X[7722], X[1112] - 3 X[12236], 3 X[381] + X[12284], 3 X[10113] - X[12292].

 

Excentral). X(11698).

 

Best regards,

Peter Moses.