HYACINTHOS 29538

[Telv Cohl]

 

Let ABC be a triangle and P a point on the Neuberg cubic K001

Denote:

Oa, Ob, Oc = the circumcenters of PBC, PCA, PAB, resp.

The reflections of AP, BP, CP in AOa, BOb, COc are concurrent.

[Antreas P. Hatzipolakis]:

Which are the points of concurrence for some points on the cubic?

 

[Peter Moses]:

Hi Antreas

X(1) -> X(1)
X(3) -> X(143)
X(4) -> X(252)

X(13) -> X(6671)
X(14) -> X(6672)
X(30) -> X(265)
X(74) -> X(186)

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X(15);

ISOGONAL CONJUGATE OF X(6671) =

 
= a^2*(Sqrt[3]*(a^4 - 3*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - 3*b^2*c^2 + c^4) - 2*(a^2 + 2*b^2 + c^2)*S)*(Sqrt[3]*(a^4 - 2*a^2*b^2 + b^4 - 3*a^2*c^2 - 3*b^2*c^2 + 2*c^4) - 2*(a^2 + b^2 + 2*c^2)*S) : :

= lies on these lines: {6, 2380}, {13, 11600}, {16, 2981}, {17, 299}, {61, 1337}, {8603, 11081}, {8742, 9112}

= isogonal conjugate of X(6671)
= isogonal conjugate of the complement of X(623)
= X(i)-isoconjugate of X(j) for these (i,j): {1, 6671}, {532, 3376}
= cevapoint of X(8603) and X(21461)
= crosssum of X(396) and X(15802)
= barycentric product X(i)*X(j) for these {i,j}: {17, 2981}, {2380, 19779}, {8603, 11119}
= barycentric quotient X(i)/X(j) for these {i,j}: {6, 6671}, {2380, 16771}, {2981, 302}, {8603, 618}, {16459, 8838}, {21461, 396}

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X(16);

 

ISOGONAL CONJUGATE OF X(6672) =  

= a^2*(Sqrt[3]*(a^4 - 3*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - 3*b^2*c^2 + c^4) + 2*(a^2 + 2*b^2 + c^2)*S)*(Sqrt[3]*(a^4 - 2*a^2*b^2 + b^4 - 3*a^2*c^2 - 3*b^2*c^2 + 2*c^4) + 2*(a^2 + b^2 + 2*c^2)*S) : :

= lies on these lines: {6, 2381}, {14, 11601}, {15, 6151}, {18, 298}, {62, 1338}, {8604, 11086}, {8741, 9113}

= isogonal conjugate of X(6672)
= isogonal conjugate of the complement of X(624)
= X(i)-isoconjugate of X(j) for these (i,j): {1, 6672}, {533, 3383}
= cevapoint of X(8604) and X(21462)
= crosssum of X(395) and X(15778)
= barycentric product X(i)*X(j) for these {i,j}: {18, 6151}, {2381, 19778}, {8604, 11120}
= barycentric quotient X(i)/X(j) for these {i,j}: {6, 6672}, {2381, 16770}, {6151, 303}, {8604, 619}, {16460, 8836}, {21462, 395}

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X(1263);

X(5)X(252)∩X(195)X(10615) = 

= (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^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 5*a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 + 6*a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 - 4*a^2*c^6 - 2*b^2*c^6 + c^8)*(3*a^12 - 12*a^10*b^2 + 19*a^8*b^4 - 16*a^6*b^6 + 9*a^4*b^8 - 4*a^2*b^10 + b^12 - 12*a^10*c^2 + 20*a^8*b^2*c^2 - 4*a^6*b^4*c^2 - 10*a^4*b^6*c^2 + 12*a^2*b^8*c^2 - 6*b^10*c^2 + 19*a^8*c^4 - 4*a^6*b^2*c^4 + 5*a^4*b^4*c^4 - 8*a^2*b^6*c^4 + 15*b^8*c^4 - 16*a^6*c^6 - 10*a^4*b^2*c^6 - 8*a^2*b^4*c^6 - 20*b^6*c^6 + 9*a^4*c^8 + 12*a^2*b^2*c^8 + 15*b^4*c^8 - 4*a^2*c^10 - 6*b^2*c^10 + c^12) : :

= lies on these lines: {5, 252}, {195, 10615}, {930, 24385}, {3459, 6150}, {3519, 27246}, {19268, 31675}

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X(3065);

 

X(1)X(15038)∩X(12)X(79) =

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

= lies on thses lines: {1, 15038}, {12, 79}, {2975, 10176}, {3336, 17483}

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Best regards,
Peter Moses.