HYACINTHOS 24151

[Antreas P. Hatzipolakis]:

 

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

Denote:

(Na), (Nb), (Nc) = the NPCs of PB'C', PC'A', PA'B', resp.

R1 = the radical axis of (Nb), (Nc)

R2 = the radical axis of (Nc), (Na)

R3 = the radical axis of (Na), (Nb)

Which is the locus of P such that :

1. the reflections of R1, R2, R3 in AA', BB', CC' are concurrent?

2. the parallels to R1, R2, R3 through A, B, C, resp. are concurrent?

H, I, G lie on the locus.

 

[César Lozada]:
 

1)      {Gibert’s quintic Q003 through ETC’s 1, 2, 4, 13, 14, 1113, 1114, 1156} \/ {q= a degree-10-circumcurve }

2)      { Lucas cubic} \/ {q}

 

Barycentric equation of q:

q= y*z*((y^2+z^2)*a^4*y^3*z^3-2*( c^2*y^2+b^2*z^2)*(a^2-b^2-c^2) *x^6+(a^4-2*a^2*b^2-2*a^2*c^2+ b^4+4*b^2*c^2+c^4)*x^6*y*z+2* a^4*y^4*z^4-2*((a^2-b^2-2*c^2) *c^2*y^3+(a^2-2*b^2-c^2)*b^2* z^3)*x^5+2*((a^4-2*a^2*b^2-3* a^2*c^2+b^4+b^2*c^2+2*c^4)*y+( a^4-3*a^2*b^2-2*a^2*c^2+2*b^4+ b^2*c^2+c^4)*z)*x^5*y*z+(7*a^ 4-6*a^2*b^2-6*a^2*c^2+2*b^4-4* b^2*c^2+2*c^4)*x^2*y^3*z^3+2*( 3*a^4-6*a^2*b^2-6*a^2*c^2+3*b^ 4-4*b^2*c^2+3*c^4)*x^4*y^2*z^ 2) + … =0

 

Points of concurrence (trilinears):

1)

Z1 (I) = 2*a^6-2*(b^2+b*c+c^2)*a^4-(2* b-c)*(b-2*c)*(b+c)^2*a^2+(2*b^ 2+b*c+2*c^2)*(b^2-c^2)^2 : :

= (R+r)*X(55)-r*X(186)

= Incircle inverse of X(6284)

= On lines:  {1,30}, {11,2072}, {12,403}, {23,3303}, {55,186}, {56,2071}, {468,612}, {497,3153}, {523,663}, {858,7191}, {1062,5433}, {1478,9642}, {1870,9629}, {2070,3295}, {3028,6000}, {3304,7464}, {3746,7575}, {3920,7426}, {4081,4511}, {4299,9641}, {5148,6020}, {5159,5272}, {5252,9577}, {5899,6767}

= [ 2.437577600592607, 2.43560628586189, 0.829439699114173 ]

 

Z1(G) = (8*a^4-11*(b^2+c^2)*a^2+14*(b^ 4-b^2*c^2+c^4))/a : :

= 7*X(2)-X(187)

= Complement of X(5215)

= On lines:  {2,187}, {620,8355}, {3788,7615}, {7617,7880}, {7622,7872}, {7817,7862}, {7848,8860}

= [ 3.374410880835517, 2.15624158367851, 0.590461440513633 ]

 

Z1(H) = (SA-24*R^2+5*SW)*SB*SC*b*c : :

= (4*R^2-SW)*X(3)+(14*R^2-3*SW)* X(4)

= Inverse of X(3515) in Circumcircle

= Inverse of X(185) in Half-Altitude circle

= Inverse of X(6353) in Orthoptic Circle of the Steiner Inellipse

= Inverse of X(20) in Polar circle

= midpoint of X(i),X(j) for these {i,j}: {4,403}

= reflection of X(i) in X(j) for these (i,j): (468,403), (2071,5159)

= On lines: {2,3}, {185,5893}, {974,1514}, {1990,6128}, {2452,10002}, {5186,5203}, {5318,8739}, {5321,8740}, {5480,8541}, {6746,10110}

= [ -3.381534618985149, -4.24213396817526, 8.138234745560395 ]

 

2)      ETC pairs (P,Z2(P)): (2, 671), (4,74), (7,1156), (8,1320), (69, 895),

 

Z2( X(20) ) = (S^2-2*SB*SC)/(SA*(S^2-3*SB* SC)*a) ::

= Trilinear pole of the line {1249,6587}

= Anticomplement of X(3184)

= reflection of X(i) in X(j) for these (i,j): (20,122), (107,4), (1304,1552), (5667,133)

= On Gibert’s cubic  K025, K447, Gibert’s curve Q001, Q107 and these lines: {2,3184}, {4,74}, {20,122}, {30,1294}, {146,648}, {253,317}, {1249,1562}, {2349,2816}, {2394,2848}, {2790,5186}, {2822,3668}, {3087,8749}, {3091,6716}, {3146,3346}

= [ -11.598596809788770, -12.73998115297866, 17.813849961564430 ]

 

César Lozada