HYACINTHOS 27024

[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.

 

 

[Randy Hutson]:

 

Dear Antreas,

1. La, Lb, Lc are concurrent at the midpoint of X(5) and X(49).

Barycentrics:

2a^10 - 7a^8(b^2 + c^2) + a^6(8b^4 + 6b^2c^2 + 8c^4) - a^4(2b^6 - b^4c^2 - b^2c^4 + 2c^6) - a^2(b^2 - c^2)^2(2b^4 + b^2c^2 + 2c^4) + (b^2 - c^2)^4(b^2 + c^2) ::
On lines {5,49}, {52,11803}, {140,9729}, {185,5498}, {550,3521}, {3530,11064}, {5972,12006}, {6143,10264}, {6689,14128}, {12134,13413}, {13366,15350}, {13383,14449}
midpoint of X(5) and X(49).

 (6,9,13) search values: (0.927263422121052, -0.933100075985801, 3.858689108918679).


2. The parallels to La, Lb, Lc through A, B, C , resp. are concurrent at X(49).

3. ABC, N1N2N3 are perspective at X(567).

4. The orthologic center (N1N2N3, ABC):

Barycentrics: 

2a^10 - 4a^8(b^2 + c^2) + a^6(b^4 + 10b^2c^2 + c^4) + a^4(b^6 + c^6) + a^2(b^2 - c^2)^2(b^4 - 7b^2c^2 + c^4) - (b^2 - c^2)^4(b^2 + c^2) : :

lies on lines {4,567}, {30,5462}, {381,12278}, {1885,13630}, {3850,13392}, {5663,12241}, {11264,12162}

(6,9,13) search values: (-4.132022417087147, -4.545852687182743, 8.694880534766500)


Best regards,
Randy Hutson