HYACINTHOS 29477

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle, A'B'C' the cevian triangle of G and A"B"C" the circumcevian triangle of G with respect A'B'C'.
The circles with diameters HA",HB",HC" intersect the NPC again at A1,B1,C1, resp.

The triangles ABC, A1B1C1 are perspective.

Perspector?

[Francisco Javier Garcia Capitan]:

It is the isotomic conjugate of SA (b^4 + c^4 - a^4) : :

Not in ETC.

 
[Antreas P. Hatzipolakis]:

Is this point  SA (b^4 + c^4 - a^4) : : now in ETC?
Also is its isotomic conjugate ?


[Peter Moses]:

Hi Antreas,

>Is this point  SA (b^4 + c^4 - a^4) : : now in ETC?

No.

X(22)X(315)∩X(25)X(317) =

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

= lies on these lines: {2, 39}, {4, 16276}, {20, 16275}, {22, 315}, {25, 317}, {69, 184}, {75, 23556}, {99, 1370}, {183, 7499}, {316, 7500}, {394, 6393}, {427, 1975}, {428, 7773}, {468, 32821}, {648, 8879}, {1007, 7392}, {1184, 7807}, {1236, 18018}, {1368, 6390}, {1369, 7492}, {1899, 12215}, {2548, 16950}, {3933, 6676}, {4159, 7737}, {4176, 4563}, {4232, 32825}, {5064, 32819}, {5094, 32820}, {5133, 11185}, {5207, 31383}, {6337, 7386}, {6340, 30786}, {6353, 32818}, {6504, 8781}, {6636, 14907}, {6995, 32816}, {6997, 7752}, {7378, 32815}, {7408, 32827}, {7409, 32826}, {7493, 7796}, {7714, 32823}, {7738, 30785}, {7776, 9909}, {7789, 11324}, {7791, 21248}, {8889, 32817}, {9723, 23195}, {10330, 33796}, {10565, 23608}, {14023, 14602}, {14033, 15437}, {14961, 28427}, {16051, 19583}, {17076, 20641}, {18138, 28738}

= isotomic conjugate of X(13854)
= isotomic conjugate of the isogonal conjugate of X(20806)
= isotomic conjugate of the polar conjugate of X(315)
= X(i)-Ceva conjugate of X(j) for these (i,j): {4590, 4611}, {18020, 4563}
= X(i)-cross conjugate of X(j) for these (i,j): {10316, 69}, {20806, 315}, {28405, 17907}
= X(i)-isoconjugate of X(j) for these (i,j): {19, 2353}, {25, 2156}, {31, 13854}, {66, 1973}, {798, 1289}, {2643, 15388}
= cevapoint of X(i) and X(j) for these (i,j): {69, 28696}, {3926, 28419}
= barycentric product X(i)*X(j) for these {i,j}: {22, 305}, {63, 20641}, {69, 315}, {76, 20806}, {127, 4590}, {304, 1760}, {345, 17076}, {670, 8673}, {1502, 10316}, {3267, 4611}, {3718, 7210}, {3926, 17907}, {4123, 7182}, {4150, 17206}, {4561, 21178}, {4563, 33294}, {4601, 18187}, {6393, 31636}
= barycentric quotient X(i)/X(j) for these {i,j}: {2, 13854}, {3, 2353}, {22, 25}, {63, 2156}, {69, 66}, {99, 1289}, {127, 115}, {206, 1974}, {249, 15388}, {305, 18018}, {315, 4}, {1370, 17407}, {1760, 19}, {1799, 16277}, {2172, 1973}, {2485, 2489}, {3313, 1843}, {3926, 14376}, {4123, 33}, {4150, 1826}, {4456, 2333}, {4463, 1824}, {4611, 112}, {5562, 27372}, {6393, 34138}, {7210, 34}, {8673, 512}, {8743, 2207}, {10316, 32}, {14396, 14398}, {16757, 6591}, {17076, 278}, {17907, 393}, {18187, 3125}, {20641, 92}, {20806, 6}, {21178, 7649}, {22075, 1501}, {23208, 27369}, {28405, 3767}, {31636, 6531}, {33294, 2501}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1180, 7803}, {2, 3926, 305}, {69, 7494, 1799}, {184, 4121, 69}, {1196, 3788, 2}

>Also is its isotomic conjugate ?  

X(13854).

Best regards,
Peter Moses.