HYACINTHOS 27027

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

Nah, Nao = the orthogonal projections of N on AH, AO, resp.

La = the Euler line of ANahNao. Similarly Lb, Lc.

N1 = the NPC center of ANahNao. Similarly N2, N3 

 

1. La, Lb, Lc are concurrent.

 

2. The parallels to La, Lb, Lc through A, B, C , resp. are concurrent.

 

3. ABC, N1N2N3 are perspective.

 

4, ABC, N1N2N3 are orthologic.

The orthologic center (ABC, N1N2N3) is the H.

 

[Peter Moses]:


Hi Antreas,

1). 2 a^10-7 a^8 b^2+8 a^6 b^4-2 a^4 b^6-2 a^2 b^8+b^10-7 a^8 c^2+6 a^6 b^2 c^2+a^4 b^4 c^2+3 a^2 b^6 c^2-3 b^8 c^2+8 a^6 c^4+a^4 b^2 c^4-2 a^2 b^4 c^4+2 b^6 c^4-2 a^4 c^6+3 a^2 b^2 c^6+2 b^4 c^6-2 a^2 c^8-3 b^2 c^8+c^10:: 

on lines {{5,49},{30,5944},{52,11803},{140,9729},{143,10096},{184,10224},{185,5498},{195,13418},{546,13403},{550,3521},{1493,12010},{1568,10610},{3530,11064},{5972,12006},{6102,10125},{6143,10264},{6689,14128},{7577,9704},{9545,10254},{10095,12242},{11558,12897},{12134,13413},{13366,15350},{13383,14449},{14940,15087}}.
midpoint of X(5) and X(49).
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8254, 10272, 5), (13434, 14643, 5).

2). X(49).

3). X(567).

4). 2 a^10-4 a^8 b^2+a^6 b^4+a^4 b^6+a^2 b^8-b^10-4 a^8 c^2+10 a^6 b^2 c^2-9 a^2 b^6 c^2+3 b^8 c^2+a^6 c^4+16 a^2 b^4 c^4-2 b^6 c^4+a^4 c^6-9 a^2 b^2 c^6-2 b^4 c^6+a^2 c^8+3 b^2 c^8-c^10:: 

9 X[381] - X[12278], X[13419] - 3 X[14893], X[11750] + 3 X[15687].

on lines {{4,567},{30,5462},{140,12897},{381,12278},{382,5422},{546,13403},{1154,13142},{1199,7728},{1885,13630},{3627,13470},{3850,13392},{5576,10113},{5663,12241},{10610,11799},{11264,12162},{11750,15687},{13419,14893}}.
Midpoint of X(i) and X(j) for these {i,j}: {{140, 12897}, {546, 13403}, {1885, 13630}, {3627, 13470}, {11264, 12162}}.


Best regards,
Peter Moses.