[Antreas P. Hatzipolakis]:
Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
La, Lb, Lc = the Euler lines of PBC, PCA, PAB, resp.
L1, L2, L3 = the reflections of La, Lb, Lc in PA', PB', PC', resp.
Li, Lii, Liii = the reflections of L1, L2, L3 in BC, CA, AB, resp.
For P = I :
1. the Li, Lii, Liii are concurrent.
2. the parallels to Li, Lii, Liii through A, B, C, resp. are concurrent.
3. the parallels to Li, Lii, Liii through A', B', C', resp. are concurrent
Points?
[Peter Moses]:
Hi Antreas,
1). a^4+2 a^3 b-2 a b^3-b^4+2 a^3 c+a^2 b c+a b^2 c+a b c^2+2 b^2 c^2-2 a c^3-c^4::
on lines {{1,5180},{2,191},{4,2771},{5,13465},{7,21},{8,79},{10,11552},{30,944},{65,5080},{78,4312},{80,4757},{145,9802},{149,3874},{320,3702},{329,442},{404,11246},{499,1749},{942,5057},{946,7701},{1046,3120},{1621,6147},{1836,3868},{2094,3652},{2478,8261},{3218,12047},{3219,12609},{3255,5558},{3303,13995},{3337,11813},{3585,4084},{3622,5426},{3647,5550},{3650,6675},{3651,5758},{3811,4338},{3873,12699},{3876,5880},{3877,10404},{3889,12701},{4018,5086},{4193,5221},{4292,4511},{4293,10052},{4654,5250},{4973,5443},{5046,5902},{5330,5434},{5499,5657},{5528,10123},{5535,6960},{5556,6598},{5603,13743},{5693,6839},{5694,6901},{5770,6841},{5884,6840},{5885,6902},{6175,11236},{6224,7354},{6763,12535},{9579,12866},{9785,10543},{9799,9812},{9965,10527},{10122,10580},{10578,11508},{12615,12937},{12773,13126}}.
anticomplement of X(191).
reflection of X(i) in X(j) for these {i,j}: {{8,2475},{21,3649},{191,11263},{442,11544},{2475,79},{3648,21},{3650,6675},{7701,946},{10266,12913},{11684,442},{12535,13089},{12937,12615},{13465,5}}.
X[8] - 4 X[79], 4 X[21] - 5 X[3616], 5 X[3616] - 2 X[3648], 5 X[3616] - 8 X[3649], X[3648] - 4 X[3649], 7 X[3622] - 6 X[5426], 8 X[3647] - 11 X[5550], 4 X[5499] - 3 X[5657], 4 X[3651] - 3 X[9778], 8 X[6841] - 9 X[9779], 8 X[442] - 7 X[9780], 3 X[2] - 4 X[11263], 15 X[3616] - 16 X[11281], 3 X[3648] - 8 X[11281], 3 X[21] - 4 X[11281], 3 X[3649] - 2 X[11281], 7 X[9780] - 16 X[11544], 7 X[9780] - 4 X[11684], 4 X[11544] - X[11684], 3 X[5603] - 2 X[13743].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (191, 11263, 2), (2895, 4647, 8), (4295, 5905, 8).
X(i)-anticomplementary conjugate of X(j) for these (i,j): {{267, 8}, {502, 1330}, {1029, 69}, {3444, 2}}.
2). X(79).
3). X(3649).
Best regards,
Peter Moses.
Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
La, Lb, Lc = the Euler lines of PBC, PCA, PAB, resp.
L1, L2, L3 = the reflections of La, Lb, Lc in PA', PB', PC', resp.
Li, Lii, Liii = the reflections of L1, L2, L3 in BC, CA, AB, resp.
For P = I :
1. the Li, Lii, Liii are concurrent.
2. the parallels to Li, Lii, Liii through A, B, C, resp. are concurrent.
3. the parallels to Li, Lii, Liii through A', B', C', resp. are concurrent
Points?
[Peter Moses]:
Hi Antreas,
1). a^4+2 a^3 b-2 a b^3-b^4+2 a^3 c+a^2 b c+a b^2 c+a b c^2+2 b^2 c^2-2 a c^3-c^4::
on lines {{1,5180},{2,191},{4,2771},{5,13465},{7,21},{8,79},{10,11552},{30,944},{65,5080},{78,4312},{80,4757},{145,9802},{149,3874},{320,3702},{329,442},{404,11246},{499,1749},{942,5057},{946,7701},{1046,3120},{1621,6147},{1836,3868},{2094,3652},{2478,8261},{3218,12047},{3219,12609},{3255,5558},{3303,13995},{3337,11813},{3585,4084},{3622,5426},{3647,5550},{3650,6675},{3651,5758},{3811,4338},{3873,12699},{3876,5880},{3877,10404},{3889,12701},{4018,5086},{4193,5221},{4292,4511},{4293,10052},{4654,5250},{4973,5443},{5046,5902},{5330,5434},{5499,5657},{5528,10123},{5535,6960},{5556,6598},{5603,13743},{5693,6839},{5694,6901},{5770,6841},{5884,6840},{5885,6902},{6175,11236},{6224,7354},{6763,12535},{9579,12866},{9785,10543},{9799,9812},{9965,10527},{10122,10580},{10578,11508},{12615,12937},{12773,13126}}.
anticomplement of X(191).
reflection of X(i) in X(j) for these {i,j}: {{8,2475},{21,3649},{191,11263},{442,11544},{2475,79},{3648,21},{3650,6675},{7701,946},{10266,12913},{11684,442},{12535,13089},{12937,12615},{13465,5}}.
X[8] - 4 X[79], 4 X[21] - 5 X[3616], 5 X[3616] - 2 X[3648], 5 X[3616] - 8 X[3649], X[3648] - 4 X[3649], 7 X[3622] - 6 X[5426], 8 X[3647] - 11 X[5550], 4 X[5499] - 3 X[5657], 4 X[3651] - 3 X[9778], 8 X[6841] - 9 X[9779], 8 X[442] - 7 X[9780], 3 X[2] - 4 X[11263], 15 X[3616] - 16 X[11281], 3 X[3648] - 8 X[11281], 3 X[21] - 4 X[11281], 3 X[3649] - 2 X[11281], 7 X[9780] - 16 X[11544], 7 X[9780] - 4 X[11684], 4 X[11544] - X[11684], 3 X[5603] - 2 X[13743].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (191, 11263, 2), (2895, 4647, 8), (4295, 5905, 8).
X(i)-anticomplementary conjugate of X(j) for these (i,j): {{267, 8}, {502, 1330}, {1029, 69}, {3444, 2}}.
2). X(79).
3). X(3649).
Best regards,
Peter Moses.
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