HYACINTHOS 27613

[Tran Quang Hung]:

 

Let ABC be a triangle.
N is the NPC center.
A'B'C' is the pedal triangle of N.

The Hatzipolakis axes of the triangles AB'C', BC'A' and CA'B' bound a triangle
A''B''C''.
Then the circumcenter of the triangle A''B''C'' lies on the Euler line of ABC.

Please, which is this point ?

 

[Angel Montesdeoca]:

*** Is the triangle center W with first barycentric coordinate:

2 a^10-5 a^8 (b^2+c^2)+2 a^6 (b^4+b^2 c^2+c^4)-a^4 (-4 b^6+b^4 c^2+b^2
c^4-4 c^6)-a^2 (b^2-c^2)^2 (4 b^4+b^2 c^2+4 c^4)+(b^2-c^2)^4 (b^2+c^2)

W is the midpoint of X(i) and X(j), for these {i, j}: {5,12107},
{26,10224}, {1658,13406}, {10020,13383}, {15331,15761}.

W is the reflection of X(i) in X(j), for these {i, j}: {5,12010},
{5498,10125}, {10125,10020} , {11250,10212}.

W lies on lines X(i)X(j) for these {i, j}: {2, 3}, {17, 11268}, {18,
11267}, {51, 8254}, {52, 11803}, {110, 3519}, {143, 12242}, {5562,
10272}, {5663, 14862}, {5944, 10619}, {5972, 10627}, {6689, 13364},
{6723, 17712}, {7691, 11805}, {8718, 15061}, {8960, 11266}, {10192,
10274}, {13561, 14864}.

(6 - 9 - 13) - search numbers of W: (0.330440219662793,
-0.539951412877810, 3.86196612790157).

Angel Montesdeoca