HYACINTHOS 26868

[Kadir Altintas]:

Let ABC be a triangle, P a point and DEF the cevian triangle of P.

Denote:

Gamma_P = the circumconic through G_P, K_P  = centroid, symmedian poin of DEF, resp.

1. For P = H,  the Gamma_H passes through K = symmedian point of ABC

 

2. G_H = X(51), K_H = X(53)
Gamma_H passes through Xi for i =216,288,343.2052.2351

Questions
-- Is this conic known?

-- More points it passes through?

 

 

[César Lozada]:

 

For P=H, the conic has the following properties:

 

a)  Barycentric equation: ∑ [ (SB-SC)*(S^2+SB*SC)*SA*a^2*y*z ] = 0

 

b) It passes through these ETC's: 6, 51, 53, 216, 288, 343, 2052, 2351, 11077, 14582

 

c) isogonal conjugate of the line {2, 95, 97, 233, 275, 317, 577, 3087, 4993, 6709, 8882, 10311, 10313, 10314, 14590, 14918}

 

d) isotomic conjugate of the line  { 76, 275, 276, 394, 458}

 

e)  Center:

Oh = SA*(SB+SC)*(2*S^2+SA^2+2*SB* SC-SW^2)*(S^2+SB*SC)*(8*(SB+ SC)*R^2-S^2-(SB+SC)^2-SW^2) : : (barycentrics)

= ((6*S^2+4*SW^2)*R^2-(S^2+SW^ 2)*SW)^2*X(3269)-16*(4*R^2-SW) *(6*SW*R^2-S^2-SW^2)*S^2*R^2* X(7668)

= on lines: {3269, 7668}

= [ 0.5267084728631982, 0.7080816533196879, 2.9073578882877350 ]

 

f) Perspector:

Qh = midpoint of X(4) and X(15412)

= SA*(SB^2-SC^2)*(S^2+SB*SC) : : (barycentrics)

= 4*X(647)-X(9409)

= lies on: Lemoine axis {187, 237}

= on lines: {4, 15412}, {130, 3269}, {137, 5099}, {187, 237}, {403, 523}, {525, 684}, {826, 3574}, {878, 10547}, {1157, 1510}, {1568, 6368}, {10254, 14592}, {14380, 14483}

= midpoint of X(4) and X(15412)

= [ -1.4233329649307620, -1.6136683849941520, 5.4147424245637590 ]

 

------------------------------ ------------------------------ ------------------------------ -------

 

For P=G,  the conic has the following properties

 

a) Equation:  ∑ [ (b^4-c^4)*y*z ] = 0

 

b) Through ETC's: 2, 66, 141, 427, 1031, 1502, 3613, 8024, 8801, 9076, 9483, 14617

 

c) isogonal conjugate of the line {6, 22}

 

d) isotomic conjugate of the line {2, 32}

 

e) Center:

Og = complement of X(4577)

= (b^4-c^4)^2 : :  (barycentrics)     very simple coordinates!!!

= on the Steiner inellipse and lines: {2, 4577}, {4, 14718}, {32, 14378}, {66, 9233}, {115, 9479}, {125, 1084}, {570, 11672}, {1031, 9483}, {2482, 6292}, {3005, 8288}, {6697, 8265}, {14416, 14424}

= midpoint of X(i) and X(j) for these {i,j}: {4, 14718}, {1031, 9483}

= complement of X(4577)

= [ 3.0418628936780470, 3.1146841905255340, 0.0804079375383724 ]

 

f) perspector = X(826)

 

 

César Lozada