POINTS ON THE McCAY CUBIC ( K003) - 1

[APH = Antreas P. Hatzipolakis]:

Let ABC be a triangle, P, Q two isogonal conjugate points and PaPbPc, QaQbQc the circumcevian triangles of P, Q, resp.

Denote

Paa = BC ∩ OPa
Pbb = CA ∩ OPb
Pcc = AB ∩ OPc

Qaa = BC ∩ OQa
Qbb = CA ∩ OQb
Qcc = AB ∩ OQc

Conjecture
Paa, Qaa, Pbb, Qbb, Pcc, Qcc lie on a conic.

For which P's the conic is a circle?

[Francisco Javier García Capitán]

It seems that there is two points other than the reflections of A, B, C in O such that the conic is a circle.

These are the intersections of the three cubics

SA SB x^2 y+SB^2 x^2 y-SA SC x^2 z-SC^2 x^2 z+2 SB^2 x y z-2 SC^2 x y z-SB SC y^2 z-SC^2 y^2 z+SB^2 y z^2+SB SC y z^2 = 0

SA^2 x y^2+SA SB x y^2-SA SC x^2 z-SC^2 x^2 z+2 SA^2 x y z-2 SC^2 x y z-SB SC y^2 z-SC^2 y^2 z+SA^2 x z^2+SA SC x z^2 = 0

SA SB x^2 y+SB^2 x^2 y-SA^2 x y^2-SA SB x y^2-2 SA^2 x y z+2 SB^2 x y z-SA^2 x z^2-SA SC x z^2+SB^2 y z^2+SB SC y z^2  = 0

ETC Search numbers for triangle 6-9-13:

{12.5795546363, 1.16658629459, -2.97292047725}, 

{0.985171152116, 10.6233155589, -4.16863297521}

 

[César Lozada]

Starting from Francisco’s result:

 

P1 [= ETC X(46357)] = 1st INTERCEPT (DISTINCT OF X(3))  OF LINE X(3)X(2574) AND McCAY CUBIC

= a*(2*OH*(OH-3*R)*S*SA*a+((S^2-3*SB*SC)*OH*b*c+(3*S^2-2*SB*SC-(18*R^2-5*SA)*SA)*S*a)*K) : :,

where K=OH*(OH-3*R)*sqrt(2*R*OH^3-S^2-SW^2-18*R^2*(3*R^2-SW))/(2*R*OH^3-S^2-SW^2-18*R^2*(3*R^2-SW))

 

= lies on cubics K003, K019, K187, K376, K443, K810, K851; curves Q007, Q008, Q009, Q010, Q020, Q063, Q113 and these lines: {3, 2574}

= reflection of P2 in X(3)

= isogonal conjugate of P2

= X(3)-vertex conjugate of-P2

= [ 12.5795546362768000, 1.1665862945942500, -2.9729204772470700 ]

 

P2 [= ETC X(46358)] = 2nd INTERCEPT (DISTINCT OF X(3))  OF LINE X(3)X(2574) AND McCAY CUBIC

= a*(2*OH*(OH-3*R)*S*SA*a-((S^2-3*SB*SC)*OH*b*c+(3*S^2-2*SB*SC-(18*R^2-5*SA)*SA)*S*a)*K) : :,

where K=OH*(OH-3*R)*sqrt(2*R*OH^3-S^2-SW^2-18*R^2*(3*R^2-SW))/(2*R*OH^3-S^2-SW^2-18*R^2*(3*R^2-SW))

 

= lies on cubics K003, K019, K187, K376, K443, K810, K851; curves Q007, Q008, Q009, Q010, Q020, Q063, Q113 and these lines: {3, 2574}

= reflection of P1 in X(3)

= isogonal conjugate of P1

= X(3)-vertex conjugate of-P1

= [ 0.9851711521159160, 10.6233155588686000, -4.1686329752088900 ]

 

[Bernard Gibert]:
 

Foci of the inconic with center O, mentioned in page K003.

Imaginary foci as well

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X(46357) = 1ST INTERCEPT, OTHER THAN X(3), OF LINE X(3)X(2574) AND McCAY CUBIC

Barycentrics    a*((2*S*b*c*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*OH-a*((b^2+c^2)*a^6-(3*b^4-2*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^4-5*b^2*c^2+3*c^4)*a^2-(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2))*K+2*S*a*(-2*(-a^2+b^2+c^2)*S*OH+3*a*b*c*(-a^2+b^2+c^2))*OH) : :
Barycentrics    a*(2*OH*(OH-3*R)*S*SA*a+((S^2-3*SB*SC)*OH*b*c+(3*S^2-2*SB*SC-(18*R^2-5*SA)*SA)*S*a)*K) : :, where K=OH*(OH-3*R)*sqrt(2*R*OH^3-S^2-SW^2-18*R^2*(3*R^2-SW))/(2*R*OH^3-S^2-SW^2-18*R^2*(3*R^2-SW))

See Antreas Hatzipolakis, Francisco Javier García Capitán, César Lozada, and Bernard Gibert, euclid 3550.

X(46357) lies on cubics K003, K019, K187, K376, K443, K810, K851; curves Q007, Q008, Q009, Q010, Q020, Q063, Q113; and this line: {3, 2574}

X(46357) = reflection of X(46358) in X(3)
X(46357) = isogonal conjugate of X(46358)
X(46357) = X(3)-vertex conjugate of-X(46358)
X(46357) = 1st focus of the inconic centered at X(3) (see cubic K003)

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X(46358) = 2ND INTERCEPT, OTHER THAN X(3), OF LINE X(3)X(2574) AND McCAY CUBIC

Barycentrics    a*(-(2*S*b*c*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*OH-a*((b^2+c^2)*a^6-(3*b^4-2*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^4-5*b^2*c^2+3*c^4)*a^2-(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2))*K+2*S*a*(-2*(-a^2+b^2+c^2)*S*OH+3*a*b*c*(-a^2+b^2+c^2))*OH) : :
Barycentrics    a*(2*OH*(OH-3*R)*S*SA*a-((S^2-3*SB*SC)*OH*b*c+(3*S^2-2*SB*SC-(18*R^2-5*SA)*SA)*S*a)*K) : :, where K=OH*(OH-3*R)*sqrt(2*R*OH^3-S^2-SW^2-18*R^2*(3*R^2-SW))/(2*R*OH^3-S^2-SW^2-18*R^2*(3*R^2-SW))

See Antreas Hatzipolakis, Francisco Javier García Capitán, César Lozada, and Bernard Gibert, euclid 3550.

X(46358) lies on cubics K003, K019, K187, K376, K443, K810, K851; curves Q007, Q008, Q009, Q010, Q020, Q063, Q113; and this line: {3, 2574}

X(46358) = reflection of X(46350) in X(3)
X(46358) = isogonal conjugate of X(46357)
X(46358) = X(3)-vertex conjugate of-X(46357)
X(46358) = 2nd focus of the inconic centered at X(3) (see cubic K003)