HYACINTHOS 29386

[Antreas P. Hatzipolakis]:

IN GENERAL

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of O,

Denote:

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

A'Na intersects again the NPC (N) at A*
B'Nb intersects again the NPC (N) at B*
C'Nc intersects again the NPC (N) at C*

A*B*C*, NaNbNc are orthologic,
The orthologic center (A*B*C*, NaNbNc) lies on the NPC (N)

[Peter Moses]:

Hi Antreas,

>The orthologic center (A*B*C*, NaNbNc) lies on the (N)  

at the midpoint of the orthocenter and the isogonal conjugate of the (intersection of the line (P, isogonal of the antigonal of P) with infinity)

(a^2 c^4 p^3 q^2+b^2 c^4 p^3 q^2-c^6 p^3 q^2+2 a^2 c^4 p^2 q^3+a^4 b^2 p^3 q r-2 a^2 b^4 p^3 q r+b^6 p^3 q r-a^4 c^2 p^3 q r-b^4 c^2 p^3 q r+2 a^2 c^4 p^3 q r+b^2 c^4 p^3 q r-c^6 p^3 q r+a^6 p^2 q^2 r-a^4 b^2 p^2 q^2 r-a^2 b^4 p^2 q^2 r+b^6 p^2 q^2 r-2 a^4 c^2 p^2 q^2 r+a^2 b^2 c^2 p^2 q^2 r-3 b^4 c^2 p^2 q^2 r+2 a^2 c^4 p^2 q^2 r+3 b^2 c^4 p^2 q^2 r-c^6 p^2 q^2 r+a^6 p q^3 r-2 a^4 b^2 p q^3 r+a^2 b^4 p q^3 r+a^4 c^2 p q^3 r-3 a^2 b^2 c^2 p q^3 r+2 a^2 c^4 p q^3 r-a^2 b^4 p^3 r^2+b^6 p^3 r^2-b^4 c^2 p^3 r^2-a^6 p^2 q r^2+2 a^4 b^2 p^2 q r^2-2 a^2 b^4 p^2 q r^2+b^6 p^2 q r^2+a^4 c^2 p^2 q r^2-a^2 b^2 c^2 p^2 q r^2-3 b^4 c^2 p^2 q r^2+a^2 c^4 p^2 q r^2+3 b^2 c^4 p^2 q r^2-c^6 p^2 q r^2+a^4 b^2 p q^2 r^2-a^2 b^4 p q^2 r^2-a^4 c^2 p q^2 r^2+a^2 c^4 p q^2 r^2+a^6 q^3 r^2-a^4 b^2 q^3 r^2+a^4 c^2 q^3 r^2-2 a^2 b^4 p^2 r^3-a^6 p q r^3-a^4 b^2 p q r^3-2 a^2 b^4 p q r^3+2 a^4 c^2 p q r^3+3 a^2 b^2 c^2 p q r^3-a^2 c^4 p q r^3-a^6 q^2 r^3-a^4 b^2 q^2 r^3+a^4 c^2 q^2 r^3) (-a^2 b^2 c^4 p^3 q^2+b^4 c^4 p^3 q^2-b^2 c^6 p^3 q^2-a^2 b^2 c^4 p^2 q^3+b^4 c^4 p^2 q^3-a^2 c^6 p^2 q^3-2 b^2 c^6 p^2 q^3+c^8 p^2 q^3-a^2 b^4 c^2 p^3 q r+b^6 c^2 p^3 q r+a^2 b^2 c^4 p^3 q r-b^2 c^6 p^3 q r-a^6 c^2 p^2 q^2 r+a^2 b^4 c^2 p^2 q^2 r+3 a^4 c^4 p^2 q^2 r+b^4 c^4 p^2 q^2 r-3 a^2 c^6 p^2 q^2 r-2 b^2 c^6 p^2 q^2 r+c^8 p^2 q^2 r-a^6 c^2 p q^3 r+2 a^2 b^4 c^2 p q^3 r-b^6 c^2 p q^3 r+a^4 c^4 p q^3 r-a^2 b^2 c^4 p q^3 r+3 b^4 c^4 p q^3 r-a^2 c^6 p q^3 r-3 b^2 c^6 p q^3 r+c^8 p q^3 r+a^2 b^4 c^2 p^3 r^2+b^6 c^2 p^3 r^2-b^4 c^4 p^3 r^2+a^6 b^2 p^2 q r^2-3 a^4 b^4 p^2 q r^2+3 a^2 b^6 p^2 q r^2-b^8 p^2 q r^2+2 b^6 c^2 p^2 q r^2-a^2 b^2 c^4 p^2 q r^2-b^4 c^4 p^2 q r^2+a^6 b^2 p q^2 r^2-3 a^4 b^4 p q^2 r^2+3 a^2 b^6 p q^2 r^2-b^8 p q^2 r^2-a^6 c^2 p q^2 r^2-3 a^2 b^4 c^2 p q^2 r^2+2 b^6 c^2 p q^2 r^2+3 a^4 c^4 p q^2 r^2+3 a^2 b^2 c^4 p q^2 r^2-3 a^2 c^6 p q^2 r^2-2 b^2 c^6 p q^2 r^2+c^8 p q^2 r^2-a^6 c^2 q^3 r^2-a^4 b^2 c^2 q^3 r^2+a^4 c^4 q^3 r^2+a^2 b^6 p^2 r^3-b^8 p^2 r^3+a^2 b^4 c^2 p^2 r^3+2 b^6 c^2 p^2 r^3-b^4 c^4 p^2 r^3+a^6 b^2 p q r^3-a^4 b^4 p q r^3+a^2 b^6 p q r^3-b^8 p q r^3+a^2 b^4 c^2 p q r^3+3 b^6 c^2 p q r^3-2 a^2 b^2 c^4 p q r^3-3 b^4 c^4 p q r^3+b^2 c^6 p q r^3+a^6 b^2 q^2 r^3-a^4 b^4 q^2 r^3+a^4 b^2 c^2 q^2 r^3) : :

Example: P = X(6) --> 

 

MIDPOINT OF X(4) AND X(6325) = 

= (b - c)^2*(b + c)^2*(2*a^6 - 2*a^4*b^2 - 2*a^2*b^4 + 2*b^6 - 2*a^4*c^2 + 3*a^2*b^2*c^2 - 5*b^4*c^2 - 2*a^2*c^4 - 5*b^2*c^4 + 2*c^6)*(5*a^6 - 2*a^4*b^2 - 5*a^2*b^4 + 2*b^6 - 2*a^4*c^2 - 2*b^4*c^2 - 5*a^2*c^4 - 2*b^2*c^4 + 2*c^6) : :

 

=  lies on the nine-point circle and these lines: {2, 6236}, {4, 6325}, {125, 32228}, {126, 549}, {381, 15922}, {542, 13234}, {690, 12494}, {1560, 6032}, {2781, 13249}, {3849, 16188}, {5099, 8704}, {9517, 12624}, {11594, 25641}

= midpoint of X(4) and X(6325)

 

= complement of X(6236)

= orthocentroidal circle inverse of X(15922)

Best regards,
Peter.

Best regards,
Peter Moses.