other VARIATION
Let ABC be a triangle and P a point.
Denote:
A', B', C' = the midpoints of AP, BP, CP, resp.
(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.
The perpendicular from A' to NaNc intersects again (Nb) at Ab
The perpendicular from A' to NaNb intersects again (Nc) at Ac
The perpendicular from B' to NbNa intersects again (Nc) at Bc
The perpendicular from B' to NbNc intersects again (Na) at Ba
The perpendicular from C' to NcNb intersects again (Na) at Ca
The perpendicular from C' to NcNa intersects again (Nb) at Cb
Ma, Mb, Mc = the midpoints of AbAc, BcBa, CaCb, resp.
Which is the locus of P such that
1. ABC, MaMbMc are orthologic?
2. The perpendicular bisectors of AbAc, BcBa, CaCb are concurrent?.
[César Lozada]:
1) Awful!
2) Locus = {sidelines} ∪ {circumcircle} ∪ {Q030 through ETC’s 1, 4, 13, 14, 74, 80} ∪ {q4: circum-quartic though X(4)}
q4: ∑[ y*z*((SB+SC)^2*y*z+2*(SA^2-S^2)*x^2) ] = 0 (barys)
For P=X(4), Ab = Ac. For P on the circumcircle Q(P) = midpoint(P, X(382))
Other ETC pairs (P,Q(P)): (1, 119), (74, 10113)
Some related centers:
Q( X(13) ) = COMPLEMENT OF X(6777)
= (2*a^6-2*(b^2+c^2)*a^4+(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2))*sqrt(3)+2*S*(2*a^4-2*(b^2+c^2)*a^2+b^4+c^4) : : (barys)
= 2*sqrt(3)*(3*SB*SC*SW+(3*SA-2*SW)*S^2)-2*S*(4*S^2-SA^2-2*SB*SC-SW^2) : : (barys)
= 3*X(13)-X(148), 3*X(14)-5*X(14061), X(99)-3*X(5464), 5*X(99)-3*X(9116), 2*X(115)-3*X(5459), 5*X(115)-3*X(31696), X(148)+3*X(617), 3*X(618)-4*X(620), 3*X(619)-2*X(620), 5*X(5459)-2*X(31696), 3*X(5460)-4*X(6722), 3*X(5464)+X(6778), 5*X(5464)-X(9116), 3*X(5478)-2*X(22515), 3*X(5613)-X(6033), 6*X(6669)-5*X(14061), X(6773)-3*X(21156), 5*X(6778)+3*X(9116), 3*X(9114)-X(20094), 3*X(15561)-X(22507)
= lies on these lines: {2, 6777}, {3, 22509}, {6, 6299}, {13, 148}, {14, 6669}, {69, 6582}, {98, 635}, {99, 299}, {114, 630}, {115, 396}, {140, 25560}, {141, 542}, {385, 533}, {532, 7813}, {621, 25236}, {623, 6771}, {624, 5613}, {633, 5982}, {2782, 25559}, {3412, 20394}, {3642, 12188}, {3643, 9862}, {5460, 6722}, {5478, 22515}, {5859, 22570}, {5981, 6115}, {6114, 7792}, {6295, 11646}, {6672, 6782}, {6770, 22687}, {6773, 21156}, {6780, 31710}, {8594, 8597}, {9113, 9760}, {9114, 20094}, {10645, 14904}, {11602, 20377}, {14136, 16529}, {15561, 22507}
= midpoint of X(i) and X(j) for these {i,j}: {3, 22509}, {13, 617}, {99, 6778}, {621, 25236}, {5859, 22570}, {5978, 22997}
= reflection of X(i) in X(j) for these (i,j): (14, 6669), (618, 619), (6782, 6672), (11602, 20377), (25560, 140)
= complement of X(6777)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5464, 6778, 99), (5613, 22689, 624)
= [ -0.3914060073780417, -4.0319972471134010, 6.6126961179296770 ]
Q( X(14) ) = COMPLEMENT OF X(6778)
= (2*a^6-2*(b^2+c^2)*a^4+(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2))*sqrt(3)-2*S*(2*a^4-2*(b^2+c^2)*a^2+b^4+c^4) : : (barys)
= 2*sqrt(3)*(3*SB*SC*SW+(3*SA-2*SW)*S^2)+2*S*(4*S^2-SA^2-2*SB*SC-SW^2) : : (barys)
= 3*X(13)-5*X(14061), 3*X(14)-X(148), X(99)-3*X(5463), 5*X(99)-3*X(9114), 2*X(115)-3*X(5460), 5*X(115)-3*X(31695), X(148)+3*X(616), 3*X(618)-2*X(620), 3*X(619)-4*X(620), 3*X(5459)-4*X(6722), 5*X(5460)-2*X(31695), 3*X(5463)+X(6777), 5*X(5463)-X(9114), 3*X(5479)-2*X(22515), 3*X(5617)-X(6033), 6*X(6670)-5*X(14061), X(6770)-3*X(21157), 5*X(6777)+3*X(9114), 3*X(9116)-X(20094), 3*X(15561)-X(22509)
= lies on these lines: {2, 6778}, {3, 22507}, {6, 6298}, {13, 6670}, {14, 148}, {69, 6295}, {98, 636}, {99, 298}, {114, 629}, {115, 395}, {140, 25559}, {141, 542}, {385, 532}, {533, 7813}, {622, 25235}, {623, 5617}, {624, 6774}, {634, 5983}, {2782, 25560}, {3411, 20395}, {3642, 9862}, {3643, 12188}, {5459, 6722}, {5479, 22515}, {5858, 22568}, {5980, 6114}, {6115, 7792}, {6582, 11646}, {6671, 6783}, {6770, 21157}, {6773, 22689}, {6779, 31709}, {8595, 8597}, {9112, 9762}, {9116, 20094}, {10646, 14905}, {11603, 20378}, {14137, 16530}, {15561, 22509}
= midpoint of X(i) and X(j) for these {i,j}: {3, 22507}, {14, 616}, {99, 6777}, {622, 25235}, {5858, 22568}, {5979, 22998}
= reflection of X(i) in X(j) for these (i,j): (13, 6670), (619, 618), (6783, 6671), (11603, 20378), (25559, 140)
= complement of X(6778)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5463, 6777, 99), (5617, 22687, 623)
= [ 6.0548239229351740, 5.9029495184967700, -3.2405269953296200 ]
Q( X(80) ) = MIDPOINT OF X(104) AND X(3436)
= (b-c)^2*a^8-2*(b+c)*(b^2+c^2)*a^7-2*(b^4+c^4-7*(b^2+c^2)*b*c)*a^6+2*(b+c)*(b^2+b*c+c^2)*(3*b^2-8*b*c+3*c^2)*a^5-4*(5*b^4+5*c^4-4*(b^2+b*c+c^2)*b*c)*b*c*a^4-2*(b^2-c^2)*(b-c)*(3*b^4+3*c^4-4*(b+c)^2*b*c)*a^3+2*(b^2-c^2)^2*(b-c)^2*(b^2+5*b*c+c^2)*a^2+2*(b^2-c^2)^3*(b-c)*(b^2-3*b*c+c^2)*a-(b^2-c^2)^4*(b-c)^2 : : (barys)
= 2*X(7681)-3*X(23513)
= lies on these lines: {3, 119}, {11, 517}, {12, 21154}, {46, 8068}, {56, 6713}, {78, 952}, {100, 6827}, {104, 3436}, {153, 6926}, {1385, 10956}, {1537, 4187}, {2478, 12775}, {2800, 21616}, {3826, 6980}, {4413, 6923}, {5533, 30323}, {5690, 18802}, {5840, 6928}, {5854, 19914}, {6174, 28459}, {6667, 7680}, {6842, 11231}, {6863, 31246}, {6948, 10728}, {6961, 11681}, {6963, 10698}, {6971, 7681}, {6978, 31272}, {7491, 24466}, {8256, 26470}, {10265, 24391}, {10955, 26287}, {10958, 26285}, {12611, 31788}, {12616, 18254}, {15017, 30503}, {15528, 21077}, {22799, 31775}
= midpoint of X(i) and X(j) for these {i,j}: {104, 3436}, {10310, 12764}
= reflection of X(i) in X(j) for these (i,j): (56, 6713), (119, 1329), (18802, 5690)
= [ 9.2383789922544230, 8.8186158164212380, -6.7283983105015950 ]
César Lozada
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