[Antreas P. Hatzipolakis]:
Let ABC be a triangle and A'B'C' the pedal triangle of H.
Denote:
Na, Nb, Nc = the NPC centers of OBC, OCA, OAB, resp.
A", B", C" = the orthogonal projections of Na, Nb, Nc on AA', BB', CC', resp.
Hi Antreas,
1) X(6143).
2) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2+a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6)::
on lines {{2,3},{6,13418},{49,11264},{ 110,5449},{112,7749},{113, 11440},{125,1614},{156,3448},{ 252,933},{562,12044},{1199, 13567},{1506,10312},{1986, 11591},{2888,11597},{3043, 5972},{3054,8744},{3580,9820}, {5876,7722},{5944,11565},{ 6152,13365},{6188,14249},{ 6247,12112},{9927,11449},{ 10104,14675},{10182,14644},{ 10540,13561},{10576,10881},{ 10577,10880},{11202,12289},{ 11447,13970},{11448,13909},{ 11464,11704},{11468,12244},{ 11562,12111},{13399,14862},{ 13450,14165}}.
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 6143), (2, 20, 6640), (2, 3549, 631), (2, 7505, 4), (2, 7558, 3525), (5, 186, 4), (5, 10018, 186), (5, 10125, 3), (24, 1656, 7577), (24, 7577, 4), (140, 403, 3520), (235, 14865, 4), (403, 3520, 4), (468, 1594, 3518), (468, 3628, 1594), (631, 3090, 6816), (858, 13383, 12088), (1594, 3518, 4), (1658, 10255, 3153), (2072, 10020, 7488), (3090, 3147, 4), (3515, 5055, 7547), (6640, 10201, 20), (9927, 11449, 12383), (14784, 14785, 10255).
orthocentroidal circle inverse of X(6143).
X(93)-Ceva conjugate of X(4).
polar conjugate of X(13585).
barycentric quotient X(4)/X(13585).
Denote:
Na, Nb, Nc = the NPC centers of OBC, OCA, OAB, resp.
A", B", C" = the orthogonal projections of Na, Nb, Nc on AA', BB', CC', resp.
L, La, Lb, Lc = the Euler lines of ABC, A"BC, B"CA, C"AB, resp.
1. L, La, Lb, Lc are concurrent.
2. The reflections of La, Lb, Lc in BC, CA, AB, resp. and L are concurrent.
1. L, La, Lb, Lc are concurrent.
2. The reflections of La, Lb, Lc in BC, CA, AB, resp. and L are concurrent.
[Peter Moses]:
Hi Antreas,
1) X(6143).
2) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2+a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6)::
on lines {{2,3},{6,13418},{49,11264},{ 110,5449},{112,7749},{113, 11440},{125,1614},{156,3448},{ 252,933},{562,12044},{1199, 13567},{1506,10312},{1986, 11591},{2888,11597},{3043, 5972},{3054,8744},{3580,9820}, {5876,7722},{5944,11565},{ 6152,13365},{6188,14249},{ 6247,12112},{9927,11449},{ 10104,14675},{10182,14644},{ 10540,13561},{10576,10881},{ 10577,10880},{11202,12289},{ 11447,13970},{11448,13909},{ 11464,11704},{11468,12244},{ 11562,12111},{13399,14862},{ 13450,14165}}.
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 6143), (2, 20, 6640), (2, 3549, 631), (2, 7505, 4), (2, 7558, 3525), (5, 186, 4), (5, 10018, 186), (5, 10125, 3), (24, 1656, 7577), (24, 7577, 4), (140, 403, 3520), (235, 14865, 4), (403, 3520, 4), (468, 1594, 3518), (468, 3628, 1594), (631, 3090, 6816), (858, 13383, 12088), (1594, 3518, 4), (1658, 10255, 3153), (2072, 10020, 7488), (3090, 3147, 4), (3515, 5055, 7547), (6640, 10201, 20), (9927, 11449, 12383), (14784, 14785, 10255).
orthocentroidal circle inverse of X(6143).
X(93)-Ceva conjugate of X(4).
polar conjugate of X(13585).
barycentric quotient X(4)/X(13585).
Best regards,
Peter Moses.
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