HYACINTHOS 26591

 

[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, the Li, Lii, Liii are concurrent. Point ?

Locus of P?

[César Lozada]

Locus : Infinite or excentral-circum-septic q7:

 

q7 = Σ[ y*z*(-(b^2-c^2)*b^2*c^2*(a^8-4 *(b^2+c^2)*a^6+(2*b^2+3*c^2)*( 3*b^2+2*c^2)*a^4-(b^2+c^2)*(4* b^4+13*b^2*c^2+4*c^4)*a^2+(b^4 +10*b^2*c^2+c^4)*(b^2-c^2)^2)* x^5+(-(2*a^10-2*(4*b^2+3*c^2)* a^8+(12*b^4+11*b^2*c^2+3*c^4)* a^6-(8*b^6-4*c^6+7*b^2*c^2*(b^ 2+3*c^2))*a^4+(b^2-c^2)*(2*b^ 6+3*c^6-b^2*c^2*(5*b^2+18*c^2) )*a^2+9*(b^2-c^2)^3*b^2*c^2)*b ^2*c^2*y+(2*a^10-2*(3*b^2+4*c^ 2)*a^8+(3*b^4+11*b^2*c^2+12*c^ 4)*a^6+(4*b^6-8*c^6-7*b^2*c^2* (3*b^2+c^2))*a^4-(b^2-c^2)*(3* b^6+2*c^6-b^2*c^2*(18*b^2+5*c^ 2))*a^2-9*(b^2-c^2)^3*b^2*c^2) *b^2*c^2*z)*x^4-(b^2-c^2)*b^2* c^2*(5*a^8-11*(b^2+c^2)*a^6+(5 *b^4+b^2*c^2+5*c^4)*a^4+3*(b^4 +b^2*c^2+c^4)*(b^2+c^2)*a^2-2* (b^2-c^2)^4)*z*y*x^3+(b^2-c^2) *a^2*(a^10-5*(b^2+c^2)*a^8+5*( 2*b^4+b^2*c^2+2*c^4)*a^6-2*(b^ 2+c^2)*(5*b^4-3*b^2*c^2+5*c^4) *a^4+(b^4+b^2*c^2+c^4)*(5*b^4- 4*b^2*c^2+5*c^4)*a^2-(b^4-c^4) *(b^2-c^2)^3)*z^2*y^2*x+(3*a^ 8-2*(4*b^2+3*c^2)*a^6+b^2*(7* b^2+4*c^2)*a^4-(b^2-c^2)*(2*b^ 4+b^2*c^2+6*c^4)*a^2+3*(b^2-c^ 2)^3*c^2)*a^4*c^2*z*y^4-((5*b^ 2-3*c^2)*a^6-(13*b^4+7*b^2*c^ 2-9*c^4)*a^4+(b^2-c^2)*(14*b^ 4+4*b^2*c^2+9*c^4)*a^2-3*(2*b^ 2+c^2)*(b^2-c^2)^3)*a^4*c^2*z^ 2*y^3-a^4*b^2*((3*b^2-5*c^2)* a^6-(9*b^4-7*b^2*c^2-13*c^4)* a^4+(b^2-c^2)*(9*b^4+4*b^2*c^ 2+14*c^4)*a^2-3*(b^2+2*c^2)*( b^2-c^2)^3)*z^3*y^2-a^4*b^2*(3 *a^8-2*(3*b^2+4*c^2)*a^6+c^2*( 4*b^2+7*c^2)*a^4+(b^2-c^2)*(6* b^4+b^2*c^2+2*c^4)*a^2-3*(b^2- c^2)^3*b^2)*z^4*y) ] = 0 (barys)

 

q7 passes through ETC’s: 1, 3, 1138

 

Intersection points Q(P):

 

Q(X(3)) = X(3)

Q(X(1138)) = X(30)

 

Q(X(1)) = X(7)X(21) ∩ X(8)X(79)

= a^4+2*(b+c)*a^3+b*c*a^2-(b+c)* (2*b^2-3*b*c+2*c^2)*a-(b^2-c^2 )^2 : : (barys)

= 3*X(2)-4*X(11263) = X(8)-4*X(79) = 4*X(21)-5*X(3616) = 3*X(21)-4*X(11281) = 8*X(442)-7*X(9780) = 5*X(3616)-2*X(3648) = 5*X(3616)-8*X(3649) = 15*X(3616)-16*X(11281) = X(3648)-4*X(3649) = 7*X(9780)-16*X(11544) = 7*X(9780)-4*X(11684) = 4*X(11544)-X(11684)

= On lines: {1, 5180}, {2, 191}, {4, 2771}, {5, 13465}, {7, 21}, {8, 79}, {10, 11552}, {30, 944}, {65, 5080}, {78, 4312}, {80, 4757}, {145, 9802}, {149, 3874}, {320, 3702}, {329, 442}, {404, 11246}, {499, 1749}, {942, 5057}, {946, 7701}, {1046, 3120}, {1621, 6147}, {1836, 3868}, {2094, 3652}, {2478, 8261}, {3218, 12047}, {3219, 12609}, {3255, 5558}, {3303, 13995}, {3337, 11813}, {3585, 4084}, {3622, 5426}, {3647, 5550}, {3650, 6675}, {3651, 5758}, {3811, 4338}, {3873, 12699}, {3876, 5880}, {3877, 10404}, {3889, 12701}, {4018, 5086}, {4193, 5221}, {4292, 4511}, {4293, 10052}, {4654, 5250}, {4973, 5443}, {5046, 5902}, {5330, 5434}, {5499, 5657}, {5528, 10123}, {5535, 6960}, {5556, 6598}, {5603, 13743}, {5693, 6839}, {5694, 6901}, {5770, 6841}, {5884, 6840}, {5885, 6902}, {6175, 11236}, {6224, 7354}, {6763, 12535}, {9579, 12866}, {9785, 10543}, {9799, 9812}, {9965, 10527}, {10122, 10580}, {10578, 11508}, {12615, 12937}, {12773, 13126}

= reflection of X(i) in X(j) for these (i,j): (8, 2475), (191, 11263), (442, 11544), (2475, 79), (3650, 6675), (10266, 12913), (11684, 442), (12535, 13089)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (7, 11415, 3616), (191, 11263, 2), (2895, 4647, 8), (3648, 3649, 3616), (4295, 5905, 8)

= [ -1.439944044879583, -0.55770736017809, 4.691359136590558 ]

 

César Lozada