HYACINTHOS 26592

 

[Antreas P.  Hatzipolakis]:

 

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

Denote:

La, Lb, Lc = the Euler lines of PBC, PCA, PAB, resp.

L1, L2, L3 = the reflections of La, Lb, Lc in PA', PB', PC', resp.

Li, Lii, Liii = the reflections of L1, L2, L3 in BC, CA, AB, resp.

For P = I :

1. the Li, Lii, Liii are concurrent.

2. the parallels to Li, Lii, Liii through A, B, C, resp. are concurrent.

3. the parallels to Li, Lii, Liii through A', B', C', resp. are concurrent 

[César Lozada]

 

1) Locus : Hyacinthos 26591

 

 

2) Locus for concurrence at Q’(P): Infinite or circumcircle or Neuberg cubic.

 

For P on Nueberg cubic,  Q’(P) lies on cubic K060.

 

ETC pairs (P,Q’(P)): (1,79), (3,4), (4,5), (13,13), (14,14), (15,11581), (16,11582), (30,5627), (74,265), (1138,30), (1263,1141), (1337,14373), (1338,14372), (3065,80), (3440,621), (3441,622), (8431,6761), (8446,11600), (8456,11601), (8487,1117), (8494,11584)

 

Q’( X(399) ) = antigonal conjugate of X(8487)

= (2*SA-3*R^2)*(S^2-9*R^2*SB+3*S B^2)*(S^2-9*R^2*SC+3*SC^2) : : (barys)

= On K060 and lines: {30, 146}, {265, 1117}, {2201, 6365}, {6761, 11584}, {10272, 14354}

= antigonal conjugate of X(8487)

= [ 1.783701445721730, -0.96330006563608, 3.484317706245479 ]

 

Q’( X(484) ) = antigonal conjugate of X(7329)

= (a^3+(b+c)*a^2-(b^2-b*c+c^2)*a -(b^2-c^2)*(b-c))/((a^2-b^2+b* c-c^2)*(a^3+(b+c)*a^2-(b^2+b* c+c^2)*a-(b^2-c^2)*(b-c))) : : (barys)

= On K060 and lines: {30, 80}, {79, 1117}

= antigonal conjugate of X(7329)

= [ -0.425557141508766, 1.04428257501176, 3.114110610672593 ]

 

3) Locus for concurrence at Q”(P): Infinite or excentral circum-septic q”:

q” = Σ [ y*z*(-(b^2-c^2)*b^2*c^2*(a^4-2 *(b^2+c^2)*a^2+10*b^2*c^2+c^4+ b^4)*x^5+(-b^2*c^2*(2*a^6-(4* b^2+c^2)*a^4+(2*b^4+8*b^2*c^2- c^4)*a^2+9*(b^2-c^2)*b^2*c^2)* y+b^2*c^2*(2*a^6-(b^2+4*c^2)* a^4-(b^4-8*b^2*c^2-2*c^4)*a^2- 9*(b^2-c^2)*b^2*c^2)*z)*x^4-2* (b^2-c^2)*b^2*c^2*(2*a^4-(b^2+ c^2)*a^2-(b^2-c^2)^2)*z*y*x^3+ 4*(b^2-c^2)*a^2*S^2*(-a^2+b^2+ c^2)*x*y^2*z^2-3*a^4*c^2*(a^2- b^2+c^2)*(c^2-a^2)*z*y^4-3*a^ 4*c^2*((2*b^2-c^2)*a^2-(b^2-c^ 2)*(2*b^2+c^2))*z^2*y^3-3*a^4* b^2*((b^2-2*c^2)*a^2-(b^2-c^2) *(b^2+2*c^2))*z^3*y^2-3*a^4*b^ 2*(a^2+b^2-c^2)*(a^2-b^2)*z^4* y)] = 0 (barys)

through ETC’s 1, 3, 1138  (Note: q” is not the same than in answer 1)

 

ETC pairs (P,Q”(P)): (1, 3649), (3, 3), (1138, 30)

 

 

César Lozada