[Antreas P. Hatzipolakis]:
Denote:
Ab, Ac = the orthogonal projections of A' on AB, AC, resp.
A2, A3 = the orthogonal projections of A' on BP, CP, resp.
Ma, M1 = the midpoints of AbAc, A2A2, resp.
Similarly Mb, M2 and Mc, M3
Mi, Mii, Miii = the midpoints of MaM1, MbM2. McM3.
1. ABC, MiMiiMiii are orthologic?
O, H lie on the locus
O, H lie on the locus
O, H lie on the locus.
For P = O, the MaM1, MbM2. McM3. are parallels to Euler line
(their perpendicular bisectors concur at the infinity point of the direction of the perpendicular to Euler line)
For P = H, the perpendicular bisectors concur at the NPC center of A'B'C'
[César Lozada]:
1) Locus = Linf ∪ {q7: circum-septic through ETC’s 3,4,5,1113,1114}
q7 =∑ [ y*z*(2*(b^2-c^2)*(-a^2+b^2+c^ 2)^2*b^2*c^2*x^5+(((3*b^2-c^2) *a^6-(5*b^4+3*b^2*c^2-3*c^4)* a^4+(b^6+7*b^4*c^2-3*c^6)*a^2+ (b^2-c^2)*(b^6-c^6-b^2*c^2*(2* b^2+c^2)))*c^2*y+((b^2-3*c^2)* a^6-(3*b^4-3*b^2*c^2-5*c^4)*a^ 4+(3*b^6-7*b^2*c^4-c^6)*a^2-( b^2-c^2)*(b^6-c^6+b^2*c^2*(b^ 2+2*c^2)))*b^2*z)*x^4+(2*a^8- 5*(b^2+c^2)*a^6+3*(b^4+4*b^2* c^2+c^4)*a^4+(b^2+c^2)*(b^4-9* b^2*c^2+c^4)*a^2-(b^6-c^6)*(b^ 2-c^2))*(b^2-c^2)*z*y*x^3+a^2* (a^6-(b^2+c^2)*a^4-((b^2-c^2)^ 2-4*b^2*c^2)*a^2+(b^4-c^4)*(b^ 2-c^2))*(b^2-c^2)*z^2*y^2*x-2* a^4*c^2*(a^2-b^2+c^2)*(a^2-c^ 2)*z*y^4+a^4*(a^6-(2*b^2-c^2)* a^4+(b^4+5*b^2*c^2-4*c^4)*a^2- 2*(b^2-c^2)*(2*b^2+c^2)*c^2)* z^2*y^3-a^4*(a^6+(b^2-2*c^2)* a^4-(4*b^4-5*b^2*c^2-c^4)*a^2+ 2*(b^2-c^2)*(b^2+2*c^2)*b^2)* z^3*y^2+2*a^4*b^2*(a^2+b^2-c^ 2)*(a^2-b^2)*z^4*y)] = 0
Some orthologic centers Q1=A->Mii, Q2=Mii->A
Q1(X(3)) = X(5)X(2883) ∩ X(53)X(1906)
= (3*S^2-16*R^2*SB+SB^2-2*SC*SA) *(3*S^2-16*R^2*SC+SC^2-2*SA* SB) : : (barys)
on lines: {5, 2883}, {53, 1906}, {3832, 17500}
= [ -5.6953736993560960, -7.1890583417472480, 11.2463388874352900 ]
Z1 = Q2(X(3)) = X(2)X(11381) ∩ X(3)X(6)
= (SB+SC)*(SA^2-16*R^2*SA-SB*SC) : : (barycentrics)
= 9*X(2)-X(11381), 21*X(2)-5*X(11439), 15*X(2)+X(12279), 7*X(3)+X(52), 3*X(3)+X(389), 13*X(3)+3*X(568), 15*X(3)+X(6243), 5*X(3)+3*X(9730), 9*X(3)-X(10625), 19*X(3)-3*X(13340), 3*X(3)-X(13348), 2*X(3)+X(15012), 5*X(3)-X(15644), 5*X(3)+X(16625), X(3)+3*X(16836), 3*X(52)-7*X(389), 15*X(52)-7*X(6243), X(52)-7*X(9729), 9*X(52)+7*X(10625), 3*X(52)+7*X(13348), 2*X(52)-7*X(15012), 5*X(52)+7*X(15644), 5*X(52)-7*X(16625), 13*X(389)-9*X(568), 5*X(389)-X(6243), X(389)-3*X(9729), 5*X(389)-9*X(9730), 3*X(389)+X(10625), 2*X(389)-3*X(15012), 5*X(389)+3*X(15644), 5*X(389)-3*X(16625), X(389)-9*X(16836), 3*X(568)-13*X(9729), 5*X(568)-13*X(9730), 6*X(568)-13*X(15012), 15*X(568)-13*X(16625), X(568)-13*X(16836), 7*X(11381)-15*X(11439), 5*X(11381)+3*X(12279)
= on lines: {2, 11381}, {3, 6}, {4, 6688}, {5, 10219}, {20, 5943}, {30, 11695}, {51, 3522}, {64, 16419}, {140, 6000}, {185, 3523}, {373, 3146}, {376, 9781}, {382, 13570}, {517, 6744}, {548, 5462}, {549, 5876}, {550, 5892}, {631, 5907}, {1204, 7485}, {1216, 13382}, {1656, 13474}, {2808, 5044}, {3090, 12045}, {3357, 7393}, {3524, 5562}, {3525, 15030}, {3526, 10575}, {3528, 15045}, {3530, 13754}, {3533, 12290}, {3628, 14915}, {3818, 6803}, {3917, 10574}, {4550, 13154}, {5054, 12162}, {5059, 11451}, {5070, 16194}, {5073, 14845}, {5446, 8703}, {5447, 12100}, {5650, 12111}, {5663, 12108}, {5889, 15692}, {5890, 10299}, {5891, 15720}, {6101, 17504}, {6102, 15606}, {6689, 15122}, {6699, 7568}, {6865, 15488}, {7496, 11440}, {8717, 13861}, {9707, 10984}, {10128, 16656}, {10170, 13491}, {10303, 15072}, {10304, 15043}, {10691, 13568}, {11812, 14128}, {12041, 13402}, {12103, 13363}, {13446, 16386}, {14831, 15698}, {15058, 15702}, {15759, 16881}
= midpoint of X(i) and X(j) for these {i,j}: {3, 9729}, {548, 5462}, {550, 10110}, {1216, 13382}, {5447, 13630}, {6102, 15606}, {13446, 16386}
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3, 389, 13348), (3, 9730, 15644), (3, 9786, 3098), (3, 13336, 11430), (3, 13347, 5092), (3, 13353, 10564), (3, 16836, 9729), (185, 3523, 3819), (389, 6243, 16625), (389, 15644, 6243), (550, 5892, 10110), (6243, 9730, 389), (9729, 13348, 389), (9729, 16625, 9730), (9730, 15644, 16625)
= [ 5.8415841905283880, 5.1790486577048490, -2.6409465229013110 ]
Q1(X(4)) = X(3); Q2(X(4)) = X(10110);
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2) Locus= {sidelines} ∪ {Linf} ∪ {circumcircle} ∪ {McCay cubic K003}
Orthologic centers Q1=A’->Mii, Q2=Mii->A’:
Q1(X(1)) =
= a^2*(a+b-c)*(a-b+c)*(b+c)*(a^ 2-b^2+3*b*c-c^2)*((b+c)*a^2-2* b*c*a-b^3-c^3) : : (barys)
= on line {1510, 4017}
= [ 15.6339469766864400, 16.7524683344941300, -15.1728660465976000 ]
Q2(X(1)) = midpoint of X(3874) and X(5795)
= 2*a^4-5*(b+c)*a^3-(b^2+6*b*c+ c^2)*a^2+5*(b^2-c^2)*(b-c)*a-( b^2-c^2)^2 : : (barys)
= 5*X(65)+3*X(3058), 3*X(65)+X(10624), 3*X(553)+X(10572), 3*X(942)-X(4298), 3*X(950)+X(1770), X(950)+3*X(5902), X(1770)-9*X(5902), 9*X(3058)-5*X(10624), X(4298)+3*X(6738), 5*X(5439)-X(6737)
= on lines: {1, 631}, {5, 12563}, {10, 15934}, {57, 4305}, {65, 3058}, {226, 10826}, {355, 5542}, {515, 942}, {516, 12433}, {517, 6744}, {519, 3812}, {553, 10572}, {938, 946}, {944, 10980}, {950, 1770}, {952, 12577}, {999, 6796}, {1159, 4301}, {1210, 11375}, {1376, 3244}, {1837, 9656}, {3241, 4917}, {3333, 4308}, {3339, 3488}, {3452, 12559}, {3487, 10175}, {3579, 15935}, {3585, 3982}, {3626, 3826}, {3634, 5719}, {3636, 6690}, {3671, 5722}, {3754, 5853}, {3874, 5795}, {4031, 4299}, {4297, 5708}, {4304, 5221}, {4311, 4860}, {5122, 15174}, {5326, 13411}, {5439, 6737}, {5587, 11036}, {5728, 5884}, {5881, 11037}, {7982, 10580}, {9578, 11518}, {9797, 11525}, {9843, 12635}, {10172, 11374}, {11019, 11373}, {11020, 15016}, {12564, 16004}
= midpoint of X(3874) and X(5795)
= {X(938), X(11529)}-Harmonic conjugate of X(946)
= [ 1.3199013978167580, 1.4453769740964900, 2.0308336239252300 ]
Q1(X(3)) = X(140)X(6000) ∩ X(570)X(1990)
= (4*S^2-(SB+SC)*(16*R^2+SA-SW)) *(S^2-(16*R^2+SA-2*SW)*SA) : : (barys)
= on lines: {140, 6000}, {570, 1990}
= [ 6.7917400384111270, 6.2238979633623420, -3.8028372027639160 ]
Q2(X(3)) = Z1
Q1(X(4)) = X(52); Q2(X(4)) = X(143)
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3)
Sorry. Too long.
César Lozada
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