[Antreas P. Hatzipolakis]:
Let ABC be a triangle and A'B'C', A"B"C" the cevian, pedal triangle of a point P, resp .
Denote:
Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.
Mab, Mac = the orthogonal projections of Ma on BB', CC', resp.
(A*B*C*, A"B"C") = a (2 a^4 b-2 a^2 b^3+2 a^4 c+2 a^3 b c-a b^3 c-b^4 c+b^3 c^2-2 a^2 c^3-a b c^3+b^2 c^3-b c^4)::
on lines {{3,5718},{11,30},{46,500},{100,524},{141,11322},{404,5241},{442,4278},{511,1155},{851,3286},{1211,13588},{1503,5078},{2352,3782},{4188,5233},{5124,7465},{5204,9840},{5347,7411},{5453,5903},{6097,13408}}.
2).
(A"B"C". A*B*C*) = X(403)
(A*B*C*, A"B"C") = 6 a^6-3 a^4 b^2-4 a^2 b^4+b^6-3 a^4 c^2+8 a^2 b^2 c^2-b^4 c^2-4 a^2 c^4-b^2 c^4+c^6::
on lines {{2,14927},{6,4232},{23,11064},{24,13568},{25,5480},{30,5972},{107,1990},{110,524},{125,468},{141,7493},{154,6353},{184,12007},{186,10117},{237,1624},{549,8717},{597,3066},{1514,10295},{1596,11202},{1620,6225},{1834,4248},{1995,3589},{2393,11746},{2883,3515},{3147,6247},{3517,12233},{3518,11745},{3524,5646},{3530,13474},{3580,14683},{3628,13419},{4226,11053},{4228,6703},{4549,14070},{5640,6329},{6000,15152},{6707,7474},{6723,11645},{7495,10546},{7575,12893},{7576,7699},{8780,11898},{9306,10154},{10272,12105},{10282,12241},{11799,12121},{11807,13392},{14002,14389}}.
Midpoint of X(i) and X(j) for these {i,j}: {{23, 11064}, {468, 1495}, {1514, 10295}, {10272, 12105}}.
X[125] - 3 X[468], X[125] + 3 X[1495], X[110] + 3 X[7426], 3 X[1514] - X[10721], 3 X[10295] + X[10721], 3 X[11799] + X[12121], 9 X[186] - X[12244], 3 X[3580] + X[14683], 5 X[7426] - X[15360], 5 X[110] + 3 X[15360].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (154, 6353, 13567), (1995, 13394, 3589).
X(3524)-line conjugate of X(5646).
Best regards,
Peter Moses.
Denote:
Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.
Mab, Mac = the orthogonal projections of Ma on BB', CC', resp.
Mbc, Mba = the orthogonal projections of Mb on CC', AA', resp.
Mca, Mcb = the orthogonal projections of Mc on AA', BB', resp.
La, Lb, Lc = the Euler lines of MaMabMac, MbMbcMba, McMcaMcb, resp.
La, Lb, Lc = the Euler lines of MaMabMac, MbMbcMba, McMcaMcb, resp.
L1, L2, L3 = the reflections of La, Lb, Lc in BC, CA, AB, resp.
A*B*C* = the triangle bounded by L1,L2, L3
1. P = I
A"B"C", A*B*C* are parallelogic.
2. P = H (then A'B'C' = A"B"C")
A"B"C", A*B*C* are parallelogic.
A"B"C", A*B*C* are parallelogic.
Parallelogic centers?
[Peter Moses]:
Hi Antreas,
1).
(A"B"C", A*B*C*) = X(1319).
Hi Antreas,
1).
(A"B"C", A*B*C*) = X(1319).
(A*B*C*, A"B"C") = a (2 a^4 b-2 a^2 b^3+2 a^4 c+2 a^3 b c-a b^3 c-b^4 c+b^3 c^2-2 a^2 c^3-a b c^3+b^2 c^3-b c^4)::
on lines {{3,5718},{11,30},{46,500},{100,524},{141,11322},{404,5241},{442,4278},{511,1155},{851,3286},{1211,13588},{1503,5078},{2352,3782},{4188,5233},{5124,7465},{5204,9840},{5347,7411},{5453,5903},{6097,13408}}.
2).
(A"B"C". A*B*C*) = X(403)
(A*B*C*, A"B"C") = 6 a^6-3 a^4 b^2-4 a^2 b^4+b^6-3 a^4 c^2+8 a^2 b^2 c^2-b^4 c^2-4 a^2 c^4-b^2 c^4+c^6::
on lines {{2,14927},{6,4232},{23,11064},{24,13568},{25,5480},{30,5972},{107,1990},{110,524},{125,468},{141,7493},{154,6353},{184,12007},{186,10117},{237,1624},{549,8717},{597,3066},{1514,10295},{1596,11202},{1620,6225},{1834,4248},{1995,3589},{2393,11746},{2883,3515},{3147,6247},{3517,12233},{3518,11745},{3524,5646},{3530,13474},{3580,14683},{3628,13419},{4226,11053},{4228,6703},{4549,14070},{5640,6329},{6000,15152},{6707,7474},{6723,11645},{7495,10546},{7575,12893},{7576,7699},{8780,11898},{9306,10154},{10272,12105},{10282,12241},{11799,12121},{11807,13392},{14002,14389}}.
Midpoint of X(i) and X(j) for these {i,j}: {{23, 11064}, {468, 1495}, {1514, 10295}, {10272, 12105}}.
X[125] - 3 X[468], X[125] + 3 X[1495], X[110] + 3 X[7426], 3 X[1514] - X[10721], 3 X[10295] + X[10721], 3 X[11799] + X[12121], 9 X[186] - X[12244], 3 X[3580] + X[14683], 5 X[7426] - X[15360], 5 X[110] + 3 X[15360].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (154, 6353, 13567), (1995, 13394, 3589).
X(3524)-line conjugate of X(5646).
Best regards,
Peter Moses.
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