HYACINTHOS 26181

[Antreas P. Hatzipolakis]:

 

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

Denote:

Bc, Cb = the orthogonal projections of B', C' on HC', HB', resp.

B3 = the reflection of Bc in B'
C2 = the reflection of Cb in C'.

M1 = the midpoint of B3C2.
Similarly M2, M3.

ABC, M1M2M3 are bilogic (perspective and orthologic)
The perspector is the G.
The orthologic centers?


[César Lozada]:

 

Orthologic centers:

 

O(M1->A) = X(389)

 

O(A->M1) = isogonal conjugate of X(578)

= 1/(a^8-3*(b^2+c^2)*a^6+(3*b^4+ 4*b^2*c^2+3*c^4)*a^4-(-c^4+b^ 4)*(b^2-c^2)*a^2-2*(b^2-c^2)^ 2*b^2*c^2) : : (barycentrics)

= 1/(cos(B-C)-cos(2*A)*cos(A)) : : (trilinears)

= On the Kiepert hyperbola and these lines: {2,389}, {3,275}, {4,216}, {5,2052}, {76,7399}, {83,7395}, {96,7592}, {98,1181}, {485,6810}, {486,6809}, {1751,7567}, {3091,8796}, {5392,13160}, {6504,6815}

= reflection of X(4) in X(8799)

= isogonal conjugate of X(578)

= [ 4.038649878010758, 6.37399993409983, -2.636096954628162 ]

 

César Lozada