HYACINTHOS 27079

[Antreas P. Hatzipolakis]:

 

 

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

Denote:

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

D = the Poncelet point of ABCP

DNa, DNb, DNc intersect the pedal circle of P (circumcircle of A'B'C') again at A*, B*, C*, resp.

Which is the locus of P such that:

1. ABC, A*B*C* are perspective ?

2. A'B'C', A*B*C* are perspective ?

[Peter Moses]:

Hi Antreas,

Locus that both ABC and A'B'C' are perspective to A*B*C* is a circular circum-quintic:

a^6 c^2 x^3 y^2-3 a^4 b^2 c^2 x^3 y^2+3 a^2 b^4 c^2 x^3 y^2-b^6 c^2 x^3 y^2-2 a^4 c^4 x^3 y^2+2 a^2 b^2 c^4 x^3 y^2+a^2 c^6 x^3 y^2+b^2 c^6 x^3 y^2+a^6 c^2 x^2 y^3-3 a^4 b^2 c^2 x^2 y^3+3 a^2 b^4 c^2 x^2 y^3-b^6 c^2 x^2 y^3-2 a^2 b^2 c^4 x^2 y^3+2 b^4 c^4 x^2 y^3-a^2 c^6 x^2 y^3-b^2 c^6 x^2 y^3+2 a^2 b^4 c^2 x^3 y z-2 b^6 c^2 x^3 y z-2 a^2 b^2 c^4 x^3 y z+2 b^2 c^6 x^3 y z+a^6 c^2 x^2 y^2 z+a^4 b^2 c^2 x^2 y^2 z-a^2 b^4 c^2 x^2 y^2 z-b^6 c^2 x^2 y^2 z-2 a^4 c^4 x^2 y^2 z+2 b^4 c^4 x^2 y^2 z+a^2 c^6 x^2 y^2 z-b^2 c^6 x^2 y^2 z+2 a^6 c^2 x y^3 z-2 a^4 b^2 c^2 x y^3 z+2 a^2 b^2 c^4 x y^3 z-2 a^2 c^6 x y^3 z-a^6 b^2 x^3 z^2+2 a^4 b^4 x^3 z^2-a^2 b^6 x^3 z^2+3 a^4 b^2 c^2 x^3 z^2-2 a^2 b^4 c^2 x^3 z^2-b^6 c^2 x^3 z^2-3 a^2 b^2 c^4 x^3 z^2+b^2 c^6 x^3 z^2-a^6 b^2 x^2 y z^2+2 a^4 b^4 x^2 y z^2-a^2 b^6 x^2 y z^2-a^4 b^2 c^2 x^2 y z^2+b^6 c^2 x^2 y z^2+a^2 b^2 c^4 x^2 y z^2-2 b^4 c^4 x^2 y z^2+b^2 c^6 x^2 y z^2+a^6 b^2 x y^2 z^2-2 a^4 b^4 x y^2 z^2+a^2 b^6 x y^2 z^2-a^6 c^2 x y^2 z^2+a^2 b^4 c^2 x y^2 z^2+2 a^4 c^4 x y^2 z^2-a^2 b^2 c^4 x y^2 z^2-a^2 c^6 x y^2 z^2+a^6 b^2 y^3 z^2-2 a^4 b^4 y^3 z^2+a^2 b^6 y^3 z^2+a^6 c^2 y^3 z^2+2 a^4 b^2 c^2 y^3 z^2-3 a^2 b^4 c^2 y^3 z^2+3 a^2 b^2 c^4 y^3 z^2-a^2 c^6 y^3 z^2-a^6 b^2 x^2 z^3+a^2 b^6 x^2 z^3+3 a^4 b^2 c^2 x^2 z^3+2 a^2 b^4 c^2 x^2 z^3+b^6 c^2 x^2 z^3-3 a^2 b^2 c^4 x^2 z^3-2 b^4 c^4 x^2 z^3+b^2 c^6 x^2 z^3-2 a^6 b^2 x y z^3+2 a^2 b^6 x y z^3+2 a^4 b^2 c^2 x y z^3-2 a^2 b^4 c^2 x y z^3-a^6 b^2 y^2 z^3+a^2 b^6 y^2 z^3-a^6 c^2 y^2 z^3-2 a^4 b^2 c^2 y^2 z^3-3 a^2 b^4 c^2 y^2 z^3+2 a^4 c^4 y^2 z^3+3 a^2 b^2 c^4 y^2 z^3-a^2 c^6 y^2 z^3==0 

through X{1,4,5,80,1113,1114,1263,2009,2010}, excentral & orthic triangles.

P = I = X(1)

(Hyacinthos 29074) 

 

 

P = N = X(5)

1)  ISOGONAl CONJUGATE OF X(27357) 

= (a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6)^2 : :

= lies on these lines : {3, 252}, {137, 1487}, {1157, 6592}, {3519, 14072}, {19268, 19552}, {27868, 31392}

 

= isogonal conjugate of X(27357) 

= {X(252),X(930)}-harmonic conjugate of X(24144)


2) X(5)X(49) ∩ X(1263)X(15038) 

 
= (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6)*(2*a^16 - 11*a^14*b^2 + 25*a^12*b^4 - 31*a^10*b^6 + 25*a^8*b^8 - 17*a^6*b^10 + 11*a^4*b^12 - 5*a^2*b^14 + b^16 - 11*a^14*c^2 + 34*a^12*b^2*c^2 - 31*a^10*b^4*c^2 - 4*a^8*b^6*c^2 + 31*a^6*b^8*c^2 - 38*a^4*b^10*c^2 + 27*a^2*b^12*c^2 - 8*b^14*c^2 + 25*a^12*c^4 - 31*a^10*b^2*c^4 - 5*a^6*b^6*c^4 + 34*a^4*b^8*c^4 - 51*a^2*b^10*c^4 + 28*b^12*c^4 - 31*a^10*c^6 - 4*a^8*b^2*c^6 - 5*a^6*b^4*c^6 - 14*a^4*b^6*c^6 + 29*a^2*b^8*c^6 - 56*b^10*c^6 + 25*a^8*c^8 + 31*a^6*b^2*c^8 + 34*a^4*b^4*c^8 + 29*a^2*b^6*c^8 + 70*b^8*c^8 - 17*a^6*c^10 - 38*a^4*b^2*c^10 - 51*a^2*b^4*c^10 - 56*b^6*c^10 + 11*a^4*c^12 + 27*a^2*b^2*c^12 + 28*b^4*c^12 - 5*a^2*c^14 - 8*b^2*c^14 + c^16) : :

= lies on these lines: {5, 49}, {1263, 15038}, {6343, 13621}, {6592, 11584}, {10096, 14071}, {14627, 31674}, {14857, 24306}
= barycentric quotient X(24385) / X(13582)


P = X(80).
1) X(8).
2) X(80).

P = X(1263).
1) X(252) 
2)  X(3)X(1141) ∩ X(140)X(10277)

 

= (a^6 - a^4*b^2 - a^2*b^4 + b^6 - 3*a^4*c^2 + a^2*b^2*c^2 - 3*b^4*c^2 + 3*a^2*c^4 + 3*b^2*c^4 - c^6)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - a^4*c^2 + a^2*b^2*c^2 + 3*b^4*c^2 - a^2*c^4 - 3*b^2*c^4 + c^6)*(2*a^10 - 7*a^8*b^2 + 8*a^6*b^4 - 2*a^4*b^6 - 2*a^2*b^8 + b^10 - 7*a^8*c^2 + 6*a^6*b^2*c^2 + 5*a^4*b^4*c^2 - a^2*b^6*c^2 - 3*b^8*c^2 + 8*a^6*c^4 + 5*a^4*b^2*c^4 + 6*a^2*b^4*c^4 + 2*b^6*c^4 - 2*a^4*c^6 - a^2*b^2*c^6 + 2*b^4*c^6 - 2*a^2*c^8 - 3*b^2*c^8 + c^10) : :

= lies on these lines: {30, 1141}, {140, 10277}
= barycentric product X(8254) and X(13582)

Best regards,
Peter Moses.