HYACINTHOS 23050

 

Antreas P. Hatzipolakis

Let ABC be a triangle.

Denote:

Na = the reflection of N in BC
Nab, Nac = the reflections of Na in NB, NC, resp.
La = the Euler line of NaNabNac
Similarly Lb, Lc

The reflections of La,Lb,Lc in BC, CA, AB, resp. are concurrent.

Locus of P?

APH

 

 [César Lozada]:

 

Let’s change N for any point P.

 

The locus of P for concurrency is the circum-(10-degree-curve) whose trilinear equation is given below, passing through ETC´s:  

3, 4, 5, 80, 265, 1113, 1114

 

Points of concurrence:

Z( O ) = Z(X(80)) =  X(3) = O

Z(H) = X(186),

Z(N) = Reflection of  (1263/54) = ( -5.571149209493124, 26.18741165618953, -11.917782414150000 )

Z( X(265) ) = ( -0.706864546807271, 14.09642617011342, -5.792154614260016 ) on line (186,2970)

Z( X(1113) ) =  (54,2575)/\(110,1113) = ( -9.293593258948066, 7.41963185716735, 2.793347007998396 )

Z( X(1114) ) =  (54,2574)/\ (110,1114) = ( 4.013397180581041, 2.43399984980022, 0.103250502546823 )

 

César Lozada 

 

Locus:

-(b^2-c^2)*b*c*a^4*u^5*v^2*(-b^2-c^2+a^2)*(-b^2+a^2-b*c-c^2)*(-b^2+a^2+b*c-c^2)+(b^2-c^2)*b*c*a^2*u^3*v^2*w^2*(-b^2-c^2+a^2)*(4*b^6-4*a^2*b^4-4*b^4*c^2-4*a^4*b^2+3*a^2*b^2*c^2-4*b^2*c^4-4*a^4*c^2-4*a^2*c^4+4*c^6+4*a^6)+(a^2-c^2)*a*c*b^2*u^2*v^3*w^2*(c^2+a^2-b^2)*(4*b^6-4*a^2*b^4-4*b^4*c^2-4*a^4*b^2+3*a^2*b^2*c^2-4*b^2*c^4-4*a^4*c^2-4*a^2*c^4+4*c^6+4*a^6)+(a^2-b^2)*a*b*c^4*v^2*w^5*(a^2+b^2-c^2)*(b^2-a*b+a^2-c^2)*(b^2+a*b+a^2-c^2)+(a^2-b^2)*a*b*c^4*u^2*w^5*(a^2+b^2-c^2)*(b^2-a*b+a^2-c^2)*(b^2+a*b+a^2-c^2)+a^3*b*(-3*a^8*b^2+2*a^4*b^6-3*a^2*b^8+9*a^6*c^2*b^2-3*b^6*c^4-3*a^8*c^2+a^6*c^4+7*a^2*b^6*c^2-2*b^8*c^2+2*c^10+b^10-6*a^2*c^8+a^10-2*b^4*c^6-5*a^4*c^4*b^2+4*c^8*b^2-11*a^4*b^4*c^2+5*a^4*c^6+7*b^4*c^4*a^2+2*a^6*b^4-5*a^2*c^6*b^2)*v^2*u^4*w+a*c^3*(c^10+7*a^6*c^2*b^2+2*b^10-2*a^4*b^6-3*a^6*b^4+9*a^2*c^6*b^2-5*b^4*c^4*a^2-2*a^8*b^2+2*a^6*c^4-5*a^2*b^6*c^2+7*a^4*b^4*c^2-11*a^4*c^4*b^2-3*a^8*c^2+b^4*c^6-3*c^8*b^2+2*a^4*c^6-3*a^2*c^8+a^10+5*b^6*c^4-6*b^8*c^2+4*a^2*b^8)*v*u^2*w^4+b^3*c*(a^4*b^6+b^10-2*a^6*c^4-5*a^4*b^4*c^2+5*a^6*b^4-6*a^8*b^2-5*a^6*c^2*b^2+9*a^2*b^6*c^2+c^10+2*b^6*c^4+7*a^2*c^6*b^2+7*a^4*c^4*b^2-11*b^4*c^4*a^2+4*a^8*c^2-2*a^2*c^8+2*a^10-3*b^8*c^2-3*a^2*b^8-3*c^8*b^2+2*b^4*c^6-3*a^4*c^6)*v^4*u*w^2+a^3*b*c^2*(-3*b^8+4*a^2*b^6+5*c^6*a^2+2*a^4*b^4-3*c^4*a^4+3*b^4*c^4-4*a^2*b^4*c^2+a^8+7*a^4*b^2*c^2-5*a^2*b^2*c^4-c^2*a^6+2*c^6*b^2-4*a^6*b^2-2*c^8)*u^4*w^3+a*b^3*c^2*(-b^8-7*a^2*b^4*c^2-2*a^4*b^4+4*a^2*b^6+3*a^8+b^6*c^2+3*b^4*c^4-3*c^4*a^4+4*a^4*b^2*c^2-4*a^6*b^2-5*c^6*b^2+5*a^2*b^2*c^4-2*c^6*a^2+2*c^8)*v^4*w^3-a*b^3*(2*a^4*b^6+b^10+9*a^2*b^6*c^2-6*c^8*b^2+2*a^6*b^4-3*b^8*c^2+4*a^2*c^8+2*c^10-5*b^4*c^4*a^2+7*a^6*c^2*b^2-3*a^8*b^2-2*a^8*c^2-3*a^2*b^8-5*a^2*c^6*b^2+a^10-2*a^4*c^6+b^6*c^4+5*b^4*c^6+7*a^4*c^4*b^2-11*a^4*b^4*c^2-3*a^6*c^4)*v^4*u^2*w-a^3*c*(-3*a^8*c^2-11*a^4*c^4*b^2-3*a^8*b^2-2*b^6*c^4+a^10-5*a^4*b^4*c^2+c^10-3*b^4*c^6-2*c^8*b^2-3*a^2*c^8+a^6*b^4+5*a^4*b^6+4*b^8*c^2-5*a^2*b^6*c^2+7*a^2*c^6*b^2+2*a^4*c^6+2*a^6*c^4-6*a^2*b^8+9*a^6*c^2*b^2+7*b^4*c^4*a^2+2*b^10)*v*u^4*w^2-b*c^3*(-11*b^4*c^4*a^2+7*a^2*b^6*c^2+5*a^6*c^4-3*a^4*b^6-2*a^2*b^8+7*a^4*b^4*c^2+c^10+2*a^10-3*c^8*b^2-2*a^6*b^4+2*b^6*c^4+9*a^2*c^6*b^2+b^10-3*a^2*c^8+2*b^4*c^6-5*a^4*c^4*b^2-3*b^8*c^2-6*a^8*c^2+4*a^8*b^2-5*a^6*c^2*b^2+a^4*c^6)*v^2*u*w^4-a^3*b^2*c*(3*b^4*c^4+2*b^6*c^2+7*a^4*b^2*c^2-5*a^2*b^4*c^2-2*b^8+5*a^2*b^6+a^8-3*c^8+2*c^4*a^4-4*a^2*b^2*c^4-3*a^4*b^4-a^6*b^2-4*c^2*a^6+4*c^6*a^2)*v^3*u^4-a*b^2*c^3*(-2*c^4*a^4+4*c^6*a^2+3*a^8-c^8+4*a^4*b^2*c^2-4*c^2*a^6+5*a^2*b^4*c^2-2*a^2*b^6-3*a^4*b^4+2*b^8-5*b^6*c^2+c^6*b^2-7*a^2*b^2*c^4+3*b^4*c^4)*v^3*w^4-a^2*b^3*c*(2*a^8+3*c^8-b^8+4*b^6*c^2-4*c^6*b^2-5*a^6*b^2-2*c^2*a^6+3*a^4*b^4+a^2*b^6-2*b^4*c^4+4*a^2*b^2*c^4-3*c^4*a^4+5*a^4*b^2*c^2-7*a^2*b^4*c^2)*v^4*u^3-(b^2-c^2)*b*c*a^4*u^5*w^2*(-b^2-c^2+a^2)*(-b^2+a^2-b*c-c^2)*(-b^2+a^2+b*c-c^2)-(a^2-c^2)*a*c*b^4*v^5*w^2*(c^2+a^2-b^2)*(-b^2+a^2-c*a+c^2)*(-b^2+a^2+c*a+c^2)-(a^2-c^2)*a*c*b^4*u^2*v^5*(c^2+a^2-b^2)*(-b^2+a^2-c*a+c^2)*(-b^2+a^2+c*a+c^2)-(a^2-b^2)*a*b*c^2*u^2*v^2*w^3*(a^2+b^2-c^2)*(4*b^6-4*a^2*b^4-4*b^4*c^2-4*a^4*b^2+3*a^2*b^2*c^2-4*b^2*c^4-4*a^4*c^2-4*a^2*c^4+4*c^6+4*a^6)+(a^2-b^2)*w*a^2*b^2*u^3*v^3*(a^8-4*a^6*b^2+2*c^2*a^6+6*a^4*b^4-4*c^4*a^4-2*a^4*b^2*c^2+4*a^2*b^2*c^4-2*c^6*a^2-4*a^2*b^6-2*a^2*b^4*c^2+3*c^8+2*b^6*c^2+b^8-4*b^4*c^4-2*c^6*b^2)+(b^2-c^2)*u*b^2*c^2*v^3*w^3*(b^8-4*b^6*c^2+2*a^2*b^6+6*b^4*c^4-4*a^4*b^4-2*a^2*b^4*c^2-2*a^6*b^2+4*a^4*b^2*c^2-4*c^6*b^2-2*a^2*b^2*c^4+2*c^6*a^2+3*a^8+c^8-4*c^4*a^4-2*c^2*a^6)-(a^2-c^2)*v*a^2*c^2*u^3*w^3*(a^8-4*c^2*a^6+2*a^6*b^2-4*a^4*b^4-2*a^4*b^2*c^2+6*c^4*a^4+4*a^2*b^4*c^2-4*c^6*a^2-2*a^2*b^6-2*a^2*b^2*c^4+3*b^8-4*b^4*c^4+c^8+2*c^6*b^2-2*b^6*c^2)+b*a^2*c^3*(-2*b^4*c^4-7*a^2*b^2*c^4-4*b^6*c^2+4*a^2*b^4*c^2+3*c^4*a^4+3*b^8+4*c^6*b^2+c^6*a^2-c^8+2*a^8-3*a^4*b^4+5*a^4*b^2*c^2-2*a^6*b^2-5*c^2*a^6)*u^3*w^4-(a^2-c^2)*u*w*b^4*v^5*(c^2+a^2-b^2)^2*(-b^2+a^2-c*a+c^2)*(-b^2+a^2+c*a+c^2)+(a^2-b^2)*u*v*c^4*w^5*(a^2+b^2-c^2)^2*(b^2-a*b+a^2-c^2)*(b^2+a*b+a^2-c^2)+(b^2-c^2)*v*w*a^4*u^5*(-b^2-c^2+a^2)^2*(-b^2+a^2-b*c-c^2)*(-b^2+a^2+b*c-c^2) = 0