HYACINTHOS 29474

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, A'B'C' the cevian triangle of G, A"B"C" the circumcevian triangle of G with respect the triangle A'B'C' and O1,O2,O3 the circumcenters of GB"C",GC"A",GA"B", resp.
The triangles ABC, O1O2O3 are perspective.

[Francisco Javier Garcia Capitan]

Generalization:

Let ABC be a triangle, A'B'C' the cevian triangle of P, A"B"C" the circumcevian triangle of P with respect the triangle A'B'C' and O1,O2,O3 the circumcenters of PB"C", PC"A", PA"B", resp.
The triangles ABC, O1O2O3 are perspective gives as locus the Yiu quintic and an octic through G and H.

The perspector for H is X381 and that for G is the isotomic conjugate of the point
{3 a^4 - 4 a^2 b^2 + b^4 - 4 a^2 c^2 - 6 b^2 c^2 + c^4 ::
 not in ETC.  

[Antreas P. Hatzipolakis]:

Questions

1. Is the point  3 a^4 - 4 a^2 b^2 + b^4 - 4 a^2 c^2 - 6 b^2 c^2 + c^4 : : now in ETC ?

2. Is its isotomic conjugate in ETC?  


 [Peter Moses]:

Hi Antreas

1) No.

 

X(2)X(6)∩X(3)X(32815) =

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

= lies on the circumconics {A,B,C,X(4),X(3815)}, {A,B,C,X(6),X(5050)}.and these lines: {2, 6}, {3, 32815}, {4, 1078}, {5, 3785}, {20, 9756}, {30, 32885}, {32, 32968}, {39, 32978}, {76, 631}, {83, 32957}, {95, 6340}, {98, 25406}, {99, 3524}, {115, 32986}, {140, 3926}, {148, 33008}, {187, 14033}, {194, 33001}, {264, 6353}, {274, 17567}, {305, 1232}, {311, 7494}, {315, 3090}, {316, 3545}, {317, 8889}, {350, 5218}, {376, 7771}, {547, 14929}, {550, 32826}, {626, 32969}, {632, 32839}, {754, 31415}, {1235, 3147}, {1272, 7664}, {1444, 16434}, {1506, 14023}, {1656, 7767}, {1799, 7392}, {1909, 7288}, {1975, 3523}, {1995, 15574}, {2548, 7780}, {2549, 32457}, {2896, 32961}, {3053, 32971}, {3091, 7750}, {3096, 32951}, {3522, 32819}, {3525, 7763}, {3526, 3933}, {3530, 32886}, {3533, 7769}, {3628, 7776}, {3734, 21843}, {3767, 4045}, {3788, 32977}, {3793, 15484}, {3934, 14001}, {3964, 16419}, {4396, 31497}, {5013, 6392}, {5023, 32981}, {5054, 6390}, {5056, 7773}, {5067, 7752}, {5071, 7811}, {5077, 16509}, {5206, 33239}, {5254, 32990}, {5286, 11285}, {5319, 6683}, {5569, 32456}, {6148, 30786}, {6194, 7616}, {6292, 33221}, {6376, 30478}, {6722, 7865}, {6781, 8182}, {6811, 12323}, {6813, 12322}, {6857, 18140}, {6910, 18135}, {7410, 10449}, {7484, 9723}, {7493, 26235}, {7622, 14148}, {7737, 32983}, {7738, 7824}, {7739, 15482}, {7745, 32987}, {7746, 7800}, {7748, 33226}, {7749, 7795}, {7751, 31401}, {7754, 31400}, {7758, 31455}, {7761, 16041}, {7762, 31404}, {7768, 32823}, {7782, 10299}, {7783, 33012}, {7784, 32972}, {7785, 32999}, {7789, 32989}, {7793, 16924}, {7799, 15709}, {7803, 32960}, {7810, 32984}, {7820, 33224}, {7823, 32962}, {7828, 32956}, {7830, 33238}, {7831, 33190}, {7832, 18840}, {7836, 33000}, {7844, 33223}, {7851, 33202}, {7854, 32976}, {7857, 14069}, {7864, 33258}, {7879, 33249}, {7885, 32963}, {7891, 33206}, {7893, 16922}, {7898, 33006}, {7899, 32958}, {7904, 14063}, {7911, 33292}, {7912, 32998}, {7919, 33230}, {7928, 33283}, {7930, 33195}, {7937, 33196}, {7938, 33248}, {7942, 33194}, {7944, 32953}, {9466, 33216}, {9769, 11061}, {10303, 32830}, {10565, 20477}, {11056, 16051}, {11167, 11172}, {11812, 32892}, {12108, 32888}, {13881, 32974}, {14061, 33285}, {14712, 33016}, {15692, 32893}, {15694, 32837}, {15702, 32833}, {15708, 32874}, {15717, 32872}, {15720, 32824}, {15721, 32869}, {16239, 32884}, {16589, 33044}, {16921, 20065}, {16925, 31276}, {17128, 32964}, {17129, 33015}, {18906, 22712}, {27269, 33055}, {31450, 32450}, {32821, 32835}

 

= anticomplement of X(31489)
= isotomic conjugate of X(14494)
= isotomic conjugate of the isogonal conjugate of X(5050)
= X(31)-isoconjugate of X(14494)
= barycentric product X(76)*X(5050)
= barycentric quotient X (i)/X(j) for these {i,j}: {2, 14494}, {5050, 6}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 69, 1007}, {2, 183, 69}, {2, 193, 3815}, {2, 385, 7736}, {2, 3620, 7778}, {2, 5232, 30761}, {2, 5304, 11174}, {2, 7610, 23055}, {2, 7735, 3618}, {2, 9740, 11163}, {2, 11160, 11184}, {2, 15589, 325}, {2, 17008, 7735}, {2, 26243, 14555}, {5, 3785, 32006}, {76, 631, 6337}, {183, 325, 15589}, {230, 11168, 15271}, {230, 15271, 2}, {325, 15589, 69}, {385, 7736, 1992}, {491, 492, 3620}, {491, 32785, 32806}, {492, 32786, 32805}, {599, 15597, 2}, {1078, 32832, 4}, {1656, 7767, 32816}, {3054, 7778, 2}, {3314, 17006, 2}, {3523, 32834, 1975}, {3526, 3933, 32829}, {3533, 32818, 7769}, {3734, 21843, 32985}, {3767, 7815, 16043}, {3785, 32838, 5}, {3815, 8667, 193}, {7610, 11168, 2}, {7610, 15271, 230}, {7746, 7800, 14064}, {7749, 7795, 32970}, {7771, 11185, 376}, {10299, 32822, 7782}, {11174, 22329, 5304}, {18840, 33189, 7832}, {21356, 23053, 2}, {32805, 32806, 3619}, {32816, 32867, 1656}

2). X(14494).

 

 

Best regards,
Peter Moses.