Let ABC be a triangle and A'B'C', A"B"C" the cevian triangles of H, G, resp.
Denote:
Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.
M1, M2, M3 = the midpoints of AA", BB", CC", resp.
Na, Nb, Nc = the NPC centers of GMaM1, GMbM2, GMcM3, resp.
N1, N2, N3 = the NPC centers of HMaM1, HMbM2, HMcM3, resp.
1. ABC, NaNbNc are orthologic.
The orthologic center (NaNbNc, ABC) lies on the Euler line.
- ABC, N1N2N3 are orthologic.
The orthologic center (N1N2N3, ABC) lies on the Euler line.
Orthologic centers?
Hi Antreas,
1) >ABC, NaNbNc are orthologic
a^2 (6 a^4-12 a^2 b^2+6 b^4+11 a^2 c^2+11 b^2 c^2-13 c^4) (6 a^4+11 a^2 b^2-13 b^4-12 a^2 c^2+11 b^2 c^2+6 c^4)::
on lines {}.
Search -4.2853574755468424869.
>The orthologic center (NaNbNc, ABC) lies on the Euler line
2 a^4+11 a^2 b^2-13 b^4+11 a^2 c^2+26 b^2 c^2-13 c^4::
on lines {{2,3},{542,6329},{1328,8981}, {3626,9955},{3629,5476},{3655, 7988},{3656,7989},{3817,5844}, {5462,11017},{6439,6561},{6440 ,6560},{10113,11694},{10592, 11238},{10593,11237}}.
Search 0.033747478921451182783.
Midpoint of X(i) and X(j) for these {i,j}: {{2, 546}, {5, 5066}, {140, 3845}, {381, 547}, {548, 3830}, {3628, 3860}, {3850, 10109}, {3853, 8703}, {10113, 11694}}.
Reflection of X(i) in X(j) for these {i,j}: {{3, 11540}, {3530, 2}, {3628, 10109}, {3845, 3856}, {3850, 5066}, {3860, 3850}, {3861, 3860}, {10109, 5}, {10124, 547}}.
13 X[2] - 5 X[3], 11 X[2] + 5 X[4], 11 X[3] + 13 X[4], X[3] - 13 X[5], X[2] - 5 X[5], X[4] + 11 X[5], 7 X[3] - 13 X[140], 7 X[2] - 5 X[140], 7 X[5] - X[140], 7 X[4] + 11 X[140], 3 X[140] - X[376], 3 X[4] - 11 X[381], 3 X[5] + X[381], 3 X[2] + 5 X[381], 3 X[140] + 7 X[381], X[376] + 7 X[381], 3 X[3] + 13 X[381], 7 X[2] + X[382], 5 X[140] + X[382], 5 X[376] + 3 X[382], 5 X[4] - 11 X[546], X[382] - 7 X[546], 5 X[381] - 3 X[546], 5 X[5] + X[546], 5 X[140] + 7 X[546], 5 X[3] + 13 X[546], 3 X[3] - 13 X[547], 3 X[140] - 7 X[547], X[376] - 7 X[547], 3 X[2] - 5 X[547], 3 X[5] - X[547], 3 X[546] + 5 X[547], 3 X[4] + 11 X[547], 19 X[3] - 13 X[548], 19 X[140] - 7 X[548], 19 X[2] - 5 X[548], 19 X[547] - 3 X[548], 19 X[5] - X[548], 19 X[381] + 3 X[548], 19 X[546] + 5 X[548], 19 X[4] + 11 X[548], 9 X[548] - 19 X[549], 9 X[3] - 13 X[549], 9 X[140] - 7 X[549], 3 X[376] - 7 X[549], 9 X[2] - 5 X[549], 9 X[5] - X[549], 3 X[547] - X[549], 3 X[381] + X[549], 9 X[546] + 5 X[549], 9 X[4] + 11 X[549], 5 X[2] - X[550], 5 X[546] + X[550], 5 X[382] + 7 X[550], 17 X[547] - 15 X[1656], 17 X[5] - 5 X[1656], 17 X[381] + 15 X[1656], 19 X[5] - 7 X[3090], X[548] - 7 X[3090], 7 X[381] - 15 X[3091], 7 X[5] + 5 X[3091], X[140] + 5 X[3091], X[376] + 15 X[3091], 7 X[547] + 15 X[3091], 7 X[1656] + 17 X[3091], 17 X[550] - 5 X[3529], 17 X[2] - X[3529], 17 X[546] + X[3529], 17 X[382] + 7 X[3529], 10 X[548] - 19 X[3530], 2 X[3529] - 17 X[3530], 10 X[3] - 13 X[3530], 10 X[549] - 9 X[3530], 10 X[140] - 7 X[3530], 2 X[550] - 5 X[3530], 10 X[547] - 3 X[3530], 10 X[5] - X[3530], 2 X[546] + X[3530], 10 X[381] + 3 X[3530], 2 X[382] + 7 X[3530], 10 X[4] + 11 X[3530], 5 X[632] - X[3534], 9 X[381] - X[3543], 9 X[547] + X[3543], 3 X[549] + X[3543], 9 X[376] + 7 X[3543], X[546] - 15 X[3545], X[381] - 9 X[3545], X[5] + 3 X[3545], X[547] + 9 X[3545], X[2] + 15 X[3545], 7 X[3528] + 5 X[3627], 4 X[548] - 19 X[3628], 4 X[3] - 13 X[3628], 4 X[549] - 9 X[3628], 4 X[140] - 7 X[3628], 4 X[2] - 5 X[3628], 2 X[3530] - 5 X[3628], 4 X[547] - 3 X[3628], 4 X[5] - X[3628], 12 X[3545] + X[3628], 4 X[381] + 3 X[3628], 4 X[546] + 5 X[3628], 4 X[4] + 11 X[3628], 19 X[4] - 11 X[3830], 19 X[546] - 5 X[3830], 19 X[381] - 3 X[3830], 19 X[5] + X[3830], 7 X[3090] + X[3830], 19 X[547] + 3 X[3830], 19 X[3628] + 4 X[3830], 19 X[2] + 5 X[3830], 19 X[140] + 7 X[3830], 19 X[549] + 9 X[3830], 19 X[3530] + 10 X[3830], 19 X[3] + 13 X[3830], 5 X[632] + 7 X[3832], X[3534] + 7 X[3832], 17 X[546] - 15 X[3839], 17 X[381] - 9 X[3839], 17 X[3545] - X[3839], 17 X[5] + 3 X[3839], 5 X[1656] + 3 X[3839], 17 X[547] + 9 X[3839], 17 X[3628] + 12 X[3839], 17 X[2] + 15 X[3839], X[3529] + 15 X[3839], 3 X[3524] + 5 X[3843], 7 X[3830] - 19 X[3845], 7 X[4] - 11 X[3845], X[382] - 5 X[3845], 7 X[546] - 5 X[3845], 7 X[381] - 3 X[3845], 5 X[3091] - X[3845], 7 X[5] + X[3845], X[376] + 3 X[3845], 7 X[547] + 3 X[3845], 7 X[3628] + 4 X[3845], 7 X[2] + 5 X[3845], 7 X[549] + 9 X[3845], 7 X[3530] + 10 X[3845], 7 X[3] + 13 X[3845], 7 X[548] + 19 X[3845], 2 X[3830] - 19 X[3850], 6 X[3839] - 17 X[3850], 2 X[4] - 11 X[3850], 10 X[3091] - 7 X[3850], 2 X[3845] - 7 X[3850], 2 X[546] - 5 X[3850], 2 X[381] - 3 X[3850], 6 X[3545] - X[3850], 2 X[5] + X[3850], X[3628] + 2 X[3850], 2 X[547] + 3 X[3850], 2 X[2] + 5 X[3850], X[3530] + 5 X[3850], 2 X[140] + 7 X[3850], 2 X[549] + 9 X[3850], 2 X[3] + 13 X[3850], 10 X[1656] + 17 X[3850], 2 X[548] + 19 X[3850], 14 X[3090] + 19 X[3850], 5 X[3850] - 14 X[3851], X[546] - 7 X[3851], 15 X[3545] - 7 X[3851], X[2] + 7 X[3851], 5 X[5] + 7 X[3851], X[3530] + 14 X[3851], 5 X[3090] + 19 X[3851], 17 X[3830] - 19 X[3853], 17 X[4] - 11 X[3853], 17 X[3845] - 7 X[3853], 17 X[546] - 5 X[3853], 17 X[381] - 3 X[3853], 17 X[3850] - 2 X[3853], 3 X[3839] - X[3853], 17 X[5] + X[3853], 5 X[1656] + X[3853], 17 X[547] + 3 X[3853], 17 X[3628] + 4 X[3853], 17 X[2] + 5 X[3853], X[3529] + 5 X[3853], 17 X[140] + 7 X[3853], 17 X[549] + 9 X[3853], 17 X[3530] + 10 X[3853], 17 X[3] + 13 X[3853], 17 X[548] + 19 X[3853], 7 X[3526] + 17 X[3854], 5 X[546] - 11 X[3855], 5 X[2] + 11 X[3855], X[550] + 11 X[3855], X[382] - 10 X[3856], 7 X[546] - 10 X[3856], 7 X[381] - 6 X[3856], 7 X[3850] - 4 X[3856], 5 X[3091] - 2 X[3856], 7 X[5] + 2 X[3856], X[140] + 2 X[3856], X[376] + 6 X[3856], 7 X[547] + 6 X[3856], 7 X[3628] + 8 X[3856], 7 X[2] + 10 X[3856], 7 X[549] + 18 X[3856], 17 X[3850] - 14 X[3857], 3 X[3839] - 7 X[3857], X[3853] - 7 X[3857], 17 X[3851] - 5 X[3857], 17 X[5] + 7 X[3857], 5 X[1656] + 7 X[3857], 17 X[3090] + 19 X[3857], 19 X[381] - 15 X[3858], 19 X[3850] - 10 X[3858], 19 X[3091] - 7 X[3858], X[3830] - 5 X[3858], 19 X[5] + 5 X[3858], X[548] + 5 X[3858], 7 X[3090] + 5 X[3858], 19 X[547] + 15 X[3858], 19 X[1656] + 17 X[3858], 13 X[3858] - 19 X[3859], 13 X[381] - 15 X[3859], 13 X[3850] - 10 X[3859], 13 X[3091] - 7 X[3859], X[3] + 5 X[3859], 13 X[5] + 5 X[3859], 13 X[547] + 15 X[3859], 13 X[1656] + 17 X[3859], 4 X[3830] - 19 X[3860], 12 X[3839] - 17 X[3860], 4 X[3853] - 17 X[3860], 4 X[4] - 11 X[3860], 4 X[3845] - 7 X[3860], 8 X[3856] - 7 X[3860], 4 X[546] - 5 X[3860], 4 X[381] - 3 X[3860], 12 X[3545] - X[3860], 4 X[5] + X[3860], 4 X[547] + 3 X[3860], 4 X[2] + 5 X[3860], 2 X[3530] + 5 X[3860], 4 X[140] + 7 X[3860], 4 X[549] + 9 X[3860], 4 X[3] + 13 X[3860], 4 X[548] + 19 X[3860], 8 X[3830] - 19 X[3861], 8 X[3853] - 17 X[3861], 8 X[4] - 11 X[3861], 8 X[3845] - 7 X[3861], 16 X[3856] - 7 X[3861], 8 X[546] - 5 X[3861], 8 X[381] - 3 X[3861], 4 X[3850] - X[3861], 8 X[5] + X[3861], 2 X[3628] + X[3861], 8 X[547] + 3 X[3861], 8 X[2] + 5 X[3861], 4 X[3530] + 5 X[3861], 8 X[140] + 7 X[3861], 8 X[549] + 9 X[3861], 8 X[3] + 13 X[3861], 8 X[548] + 19 X[3861], 7 X[3528] - 15 X[5054], X[3627] + 3 X[5054], 7 X[2] - 15 X[5055], 7 X[3628] - 12 X[5055], X[376] - 9 X[5055], 7 X[547] - 9 X[5055], 7 X[5] - 3 X[5055], X[140] - 3 X[5055], 7 X[3545] + X[5055], 5 X[3091] + 3 X[5055], X[3845] + 3 X[5055], 2 X[3856] + 3 X[5055], 7 X[3850] + 6 X[5055], 7 X[381] + 9 X[5055], 7 X[3860] + 12 X[5055], X[382] + 15 X[5055], 7 X[546] + 15 X[5055], 7 X[3839] + 17 X[5055], 3 X[5054] - 11 X[5056], X[3627] + 11 X[5056], 19 X[376] - 3 X[5059], 19 X[140] - X[5059], 7 X[548] - X[5059], 7 X[3830] + X[5059], 19 X[3845] + X[5059], 19 X[382] + 5 X[5059], X[3830] - 19 X[5066], 5 X[3858] - 19 X[5066], 3 X[3839] - 17 X[5066], X[3853] - 17 X[5066], 7 X[3857] - 17 X[5066], 5 X[3859] - 13 X[5066], X[4] - 11 X[5066], X[3861] - 8 X[5066], 5 X[3091] - 7 X[5066], X[3845] - 7 X[5066], 2 X[3856] - 7 X[5066], X[546] - 5 X[5066], 7 X[3851] - 5 X[5066], X[3860] - 4 X[5066], X[381] - 3 X[5066], 3 X[3545] - X[5066], X[547] + 3 X[5066], X[3628] + 4 X[5066], X[2] + 5 X[5066], X[140] + 7 X[5066], 3 X[5055] + 7 X[5066], X[549] + 9 X[5066], X[3530] + 10 X[5066], X[3] + 13 X[5066], 5 X[1656] + 17 X[5066], X[548] + 19 X[5066], 7 X[3090] + 19 X[5066], 7 X[140] - 13 X[5067], 7 X[3845] + 13 X[5067], 14 X[3856] + 13 X[5067], X[5] - 13 X[5068], 3 X[3545] + 13 X[5068], X[5066] + 13 X[5068], 9 X[1656] - 17 X[5071], 9 X[5] - 5 X[5071], 3 X[547] - 5 X[5071], X[549] - 5 X[5071], 3 X[381] + 5 X[5071], 9 X[5066] + 5 X[5071], 9 X[3091] + 7 X[5071], 9 X[3850] + 10 X[5071], 9 X[3859] + 13 X[5071], X[3543] + 15 X[5071], 9 X[3858] + 19 X[5071], 3 X[3545] - 11 X[5072], X[5066] - 11 X[5072], X[5] + 11 X[5072], 13 X[5068] + 11 X[5072], 5 X[2] - 13 X[5079], X[550] - 13 X[5079], 5 X[546] + 13 X[5079], 11 X[3855] + 13 X[5079], X[3629] - 5 X[5476], X[3655] - 9 X[7988], X[3656] + 7 X[7989], 17 X[548] - 19 X[8703], 17 X[3] - 13 X[8703], 17 X[3530] - 10 X[8703], 17 X[549] - 9 X[8703], 17 X[140] - 7 X[8703], 17 X[2] - 5 X[8703], X[3529] - 5 X[8703], 17 X[3628] - 4 X[8703], 17 X[547] - 3 X[8703], 17 X[5] - X[8703], 5 X[1656] - X[8703], 3 X[3839] + X[8703], 7 X[3857] + X[8703], 17 X[5066] + X[8703], 17 X[3850] + 2 X[8703], 17 X[381] + 3 X[8703], 17 X[3860] + 4 X[8703], 17 X[546] + 5 X[8703], 17 X[3845] + 7 X[8703], 17 X[3861] + 8 X[8703], 17 X[4] + 11 X[8703], 17 X[3830] + 19 X[8703], X[3626] + 5 X[9955], 2 X[548] - 19 X[10109], 14 X[3090] - 19 X[10109], 10 X[1656] - 17 X[10109], 2 X[8703] - 17 X[10109], 2 X[3] - 13 X[10109], 2 X[549] - 9 X[10109], 10 X[5071] - 9 X[10109], 2 X[140] - 7 X[10109], 6 X[5055] - 7 X[10109], 2 X[2] - 5 X[10109], X[3530] - 5 X[10109], 2 X[547] - 3 X[10109], 6 X[3545] + X[10109], 2 X[5066] + X[10109], X[3860] + 2 X[10109], 2 X[381] + 3 X[10109], X[3861] + 4 X[10109], 2 X[546] + 5 X[10109], 14 X[3851] + 5 X[10109], 10 X[3091] + 7 X[10109], 2 X[3845] + 7 X[10109], 4 X[3856] + 7 X[10109], 2 X[4] + 11 X[10109], 10 X[3859] + 13 X[10109], 6 X[3839] + 17 X[10109], 2 X[3853] + 17 X[10109], 14 X[3857] + 17 X[10109], 2 X[3830] + 19 X[10109], 10 X[3858] + 19 X[10109], 6 X[548] - 19 X[10124], 6 X[8703] - 17 X[10124], 6 X[3] - 13 X[10124], 6 X[140] - 7 X[10124], 2 X[376] - 7 X[10124], 18 X[5055] - 7 X[10124], 6 X[2] - 5 X[10124], 3 X[3530] - 5 X[10124], 2 X[549] - 3 X[10124], 10 X[5071] - 3 X[10124], 3 X[3628] - 2 X[10124], 6 X[5] - X[10124], 3 X[10109] - X[10124], 2 X[381] + X[10124], 18 X[3545] + X[10124], 3 X[3850] + X[10124], 6 X[5066] + X[10124], 3 X[3860] + 2 X[10124], 3 X[3861] + 4 X[10124], 6 X[546] + 5 X[10124], 6 X[3845] + 7 X[10124], 12 X[3856] + 7 X[10124], 2 X[3543] + 9 X[10124], 6 X[4] + 11 X[10124], 18 X[3839] + 17 X[10124], 6 X[3853] + 17 X[10124], 6 X[3830] + 19 X[10124], 5 X[3534] - 13 X[10299], 11 X[5070] - 3 X[10304], 17 X[7486] - X[11001], X[5462] + 2 X[11017], 19 X[10124] - 18 X[11539], 19 X[2] - 15 X[11539], 19 X[3628] - 12 X[11539], 19 X[547] - 9 X[11539], 19 X[5055] - 7 X[11539], 19 X[10109] - 6 X[11539], 19 X[5] - 3 X[11539], X[548] - 3 X[11539], 7 X[3090] - 3 X[11539], 19 X[3545] + X[11539], X[3830] + 3 X[11539], 5 X[3858] + 3 X[11539], 19 X[5066] + 3 X[11539], 19 X[3850] + 6 X[11539], 19 X[381] + 9 X[11539], 19 X[3860] + 12 X[11539], 19 X[546] + 15 X[11539], 19 X[3839] + 17 X[11539], 13 X[549] - 18 X[11540], 13 X[140] - 14 X[11540], 13 X[10124] - 12 X[11540], 13 X[2] - 10 X[11540], 13 X[3628] - 8 X[11540], 13 X[547] - 6 X[11540], 13 X[10109] - 4 X[11540], 13 X[5] - 2 X[11540], 5 X[3859] + 2 X[11540], 13 X[5066] + 2 X[11540], 13 X[3850] + 4 X[11540], 13 X[381] + 6 X[11540], 13 X[3856] + 7 X[11540], 13 X[3860] + 8 X[11540], 13 X[546] + 10 X[11540], 13 X[3845] + 14 X[11540], 13 X[3861] + 16 X[11540], 7 X[3534] + X[11541], X[10989] + 3 X[11563].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3528,5054),(2,3545,3851),(2 ,3839,3529),(5,381,547),(5,549 ,5071),(5,550,5079),(5,3091, 140),(5,3545,5066),(5,3627, 5056),(5,3845,5055),(5,3850, 3628),(5,3851,546),(5,3857, 1656),(5,3858,3090),(140,546, 382),(140,3091,3856),(376,381, 3845),(376,3091,381),(381,5055 ,376),(381,5071,549),(382,3851 ,3091),(382,5055,2),(547,5066, 381),(547,10124,3628),(549, 5071,547),(550,3855,546),(1656 ,3839,8703),(1656,3857,3853),( 3090,3830,11539),(3090,3858, 548),(3091,3856,3850),(3091, 5055,3845),(3529,3857,546),( 3530,3850,546),(3628,3850, 3861),(3830,11539,548),(3839, 8703,3853),(3845,5055,140),( 3845,11539,5059),(3851,5079, 3855),(3855,5079,550),(3857, 8703,3839),(3858,11539,3830),( 5055,5066,3856),(5066,10109, 3860),(5068,5072,5),(10109, 10124,547)
------------------------------ ------------------------------ ------------------------------ -------
2) >ABC, N1N2N3 are orthologic
a^2 (2 a^4-4 a^2 b^2+2 b^4+5 a^2 c^2+5 b^2 c^2-7 c^4) (2 a^4+5 a^2 b^2-7 b^4-4 a^2 c^2+5 b^2 c^2+2 c^4):: g X(3534)
on lines {{3,7712},{69,11001},{265,3543 },{2777,11564},{3431,6000},{ 3519,5059},{3521,3832},{3545, 4846},{6413,6480},{6414,6481}} .
9 X[3545] - 10 X[7703], 6 X[3] - 5 X[7712].
Search 72.930498921161258591.
on Jerabek & K330.
X(11455)-cross conjugate of X(4).
vertex conjugate of X(j) and X(j) for these (i,j): {{3,3431},{2163,10623},{3426,1 1270},{7612,11181}}.
>The orthologic center (N1N2N3, ABC) lies on the Euler line
X(3861).
Best regards,
Peter Moses.
[César Lozada]:
Centers (all coordinates are trilinears):
1) Z1(A->Na) = a/(13*a^4-11*(b^2+c^2)*a^2-6*( b^2-c^2)^2) : :
= [ -4.285357475546842, -4.06272509342886, 8.431177612225978 ]
= isogonal conjugate of {3,3054}/\{6,1657}
Z(Na->A) = (2*a^4+11*(b^2+c^2)*a^2-13*(b^ 2-c^2)^2)/a : :
= 13*cos(B-C)-2*cos(A) : :
= 11*X(3)+13*X(4)
= Shinagawa coefficients (11,15)
= On lines: {2,3}, {542,6329}, {1328,8981}, {3626,9955}, {3629,5476}, {3655,7988}, {3656,7989}, {3817,5844}, {5462,11017}, {6439,6561}, {6440,6560}, {10592,11238}, {10593,11237}
= midpoint of X(i) and X(j) for these {i,j}: {2,546}, {5,5066}, {140,3845}, {381,547}, {548,3830}, {3628,3860}, {3850,10109}, {3853,8703}
= reflection of X(i) in X(j) for these (i,j): (3,11540), (3530,2), (3628,10109), (3845,3856), (3850,5066), (3860,3850), (3861,3860), (10109,5), (10124,547)
= [ 0.033747478921451, -0.83586147067768, 4.203762048259028 ]
2) Z2(A->N1) = a/(7*a^4-5*(b^2+c^2)*a^2-2*(b^ 2-c^2)^2) : :
= 1/(2*cos(B-C)-7*cos(A)) : :
= On Jerabek hyperbola, cubic K330 and these lines: {3,7712}, {69,11001}, {265,3543}, {2777,11564}, {3431,6000}, {3519,5059}, {3521,3832}, {3545,4846}, {6413,6480}, {6414,6481}
= isogonal conjugate of X(3534)
= [ 72.930498921161260, 77.60764665262488, -83.747936548907280 ]
Z2(N1->A) = X(3861), on the Euler line
César Lozada
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