HYACINTHOS 25355

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, A'B'C' the pedal triangle of H and L a line passing through H.

Denote:

A', B", C" = the orthogonal projections of A', B', C' on L, resp.

Ab, Ac = the orthogonal projections of A" on HB, HC, resp.

Bc, Ba = the orthogonal projections of B" on HC, HA, resp.

Ca, Cb = the orthogonal projections of C" on HA, HB, resp.

Na, Nb, Nc = the NPC centers of A"AbAc, B"BcBa, C"CaCb, resp.

A'B'C', NaNbNc are orthologic.

1. Which are the orthologic centers for L = Euler line?

2. Which are the loci of the orthologic centers as L moves around H?

 

[César Lozada]:

 

 

From general to particular:

 

2) Zn = (NaNbNc- > A’B’C’) lies on a conic (it is always an ellipse) with center:

 

On =  (SB+SC)*(5*SA^2-(14*R^2+3*SW) *SA+8*S^2) : : (barycentrics)

= on lines: {4,52}, {51,1657}, {140,6688}, {511,3850}, {550,5462}, {1216,3851}, {1656,5447}, {3522,5892}, {3523,11465}, {3854,5891}, {3858,10263}, {5056,10625}, {5059,9730}, {5068,10170}, {10219,11592}, {10575,11002}

= [ -3.826837157509331, -4.01907787735875, 8.189335546621663 ]

 

and perspector Pn not related to any ETC center.

Pn = a^2/(7*a^12-55*(b^2+c^2)*a^10+ (130*b^4+99*b^2*c^2+130*c^4)* a^8-2*(b^2+c^2)*(65*b^4-42*b^ 2*c^2+65*c^4)*a^6+5*(b^2-c^2)^ 2*(11*b^4+4*b^2*c^2+11*c^4)*a^ 4-(b^4-c^4)*(b^2-c^2)*(7*b^4- 94*b^2*c^2+7*c^4)*a^2-9*(b^2- c^2)^4*b^2*c^2) : : (barycentrics)

 

The only ETC center on this conic is X(1112).

 

Za= (A’B’C’ -> NaNbNc)  lies on a very nice curve (magenta-colored in the attached figure). Sorry, I can’t add anymore.

 

1)      For L=Euler line of ABC

Zn = (NaNbNc- > A’B’C’) = X(1112)

Za = (A’B’C’ -> NaNbNc) =

((4*cos(2*A)+5)*cos(B-C)+cos( A)*cos(2*(B-C))-9*cos(A)-cos( 3*A))*((cos(2*A)+5/2)*cos(B-C) -2*cos(A)+cos(3*A))*sec(A) : : (tril.)

= on line {6000, 10295}

= [ -1.326249460991653, -3.15594148171695, 6.437662182015337 ]

 

César Lozada