[Antreas P. Hatzipolakis]:
Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
Ab, Ac = the orthogonal projections of A' on AC, AB, resp.
Abc, Acb = the orthogonal projections of Ab, Ac on AB, AC, resp.
A* = AbAbc ∩ AcAcb,
Similarly B*, C*.
Which is the locus of P such that
1. ABC, A*B*C* are perspective?
I lies on the locus. Perspector = X(55)
2. ABC, A*B*C* are orthologic?
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[Ercole Suppa]
(1) locus locus of P such that ABC, A*B*C* are perspective = Darboux cubic K004 through X(i) for these i: {1,3,4,20,40,64,84,1490,1498,2130,2131,3182,3183,3345,3346,3347,3348,3353,3354,3355,3472,3473,3637}
*** pairs {P = X(i),Q = X(j)} for these {i,j}: {1,55},{3,6},{4,3},{20,25},{40,56},{64,154},{84,198},{1490,1436},{1498,64},{3182,7037},{3345,1035},{3346,1033}
*** some points Q = Q(X(i))
Q1 = Q(X(2130)) = ISOGONAL CONJUGATE OF X(14362)
= a^2 (3 a^4-(b^2-c^2)^2-2 a^2 (b^2+c^2)) (a^12-6 a^10 (b^2-c^2)-2 a^2 (b^2-c^2)^4 (3 b^2+5 c^2)+a^8 (15 b^4+14 b^2 c^2-29 c^4)+(b^2-c^2)^4 (b^4+10 b^2 c^2+5 c^4)-4 a^6 (5 b^6+5 b^4 c^2-b^2 c^4-9 c^6)+a^4 (15 b^8-20 b^6 c^2+50 b^4 c^4-36 b^2 c^6-9 c^8)) (a^12+6 a^10 (b^2-c^2)-2 a^2 (b^2-c^2)^4 (5 b^2+3 c^2)+(b^2-c^2)^4 (5 b^4+10 b^2 c^2+c^4)+a^8 (-29 b^4+14 b^2 c^2+15 c^4)+4 a^6 (9 b^6+b^4 c^2-5 b^2 c^4-5 c^6)+a^4 (-9 b^8-36 b^6 c^2+50 b^4 c^4-20 b^2 c^6+15 c^8)) : : (barys)
= lies on these lines: {3,2130}, {154,1033}, {2060,14365}
= isogonal conjugate of X(14362)
=ETC 6-9-13 search numbers [-0.0652824214609015903, -0.171691718261267074, 3.78965832753182627]
Q2 = Q(X(2131)) = X(3)X(2131) ∩ X(1436)X(7037)
= -a^2 (a^2-b^2-c^2) (a^8-4 a^6 (b^2-c^2)+(b^2-c^2)^4-4 a^2 (b^2-c^2) (b^2+c^2)^2+2 a^4 (3 b^4+2 b^2 c^2-5 c^4)) (a^8+4 a^6 (b^2-c^2)+(b^2-c^2)^4+4 a^2 (b^2-c^2) (b^2+c^2)^2+a^4 (-10 b^4+4 b^2 c^2+6 c^4)) (a^16-8 a^14 (b^2+c^2)-56 a^10 (b^2-c^2)^2 (b^2+c^2)-8 a^2 (b^2-c^2)^6 (b^2+c^2)+(b^2-c^2)^6 (b^4+14 b^2 c^2+c^4)+4 a^12 (7 b^4-10 b^2 c^2+7 c^4)+2 a^8 (b^2-c^2)^2 (35 b^4+114 b^2 c^2+35 c^4)-8 a^6 (b^2-c^2)^2 (7 b^6+25 b^4 c^2+25 b^2 c^4+7 c^6)+4 a^4 (b^2-c^2)^2 (7 b^8+50 b^4 c^4+7 c^8)) : : (barys)
= lies on these lines: {3,2131}, {1436,7037}
=ETC 6-9-13 search numbers [-4.35775772132550380, -6.90963960045842761, 10.4355339228366000]
Q3 = Q(X(3183)) = ISOGONAL CONJUGATE OF X(14361)
= a^2 (a^2-b^2-c^2) (a^8-4 a^6 (b^2-c^2)+(b^2-c^2)^4-4 a^2 (b^2-c^2) (b^2+c^2)^2+2 a^4 (3 b^4+2 b^2 c^2-5 c^4)) (a^8+4 a^6 (b^2-c^2)+(b^2-c^2)^4+4 a^2 (b^2-c^2) (b^2+c^2)^2+a^4 (-10 b^4+4 b^2 c^2+6 c^4)) : : (barys)
= S^4 + (96 R^4-SB SC-32 R^2 SW+2 SW^2) S^2 + 256 R^6 SB+256 R^6 SC-64 R^4 SB SC-128 R^4 SB SW-128 R^4 SC SW+24 R^2 SB SC SW+16 R^2 SB SW^2+16 R^2 SC SW^2-2 SB SC SW^2 : :
= lies on these lines: {3,1033}, {6,14092}, {154,577}, {198,1035}, {1032,3964}, {1092,15905}, {1105,20792},
{1598,13855}, {15341,16391}
= isogonal conjugate of X(14361)
=ETC 6-9-13 search numbers [1.82861473596034199, 2.00978721618410853, 1.40529730025983843]
Q4 = Q(X(3347)) = X(3)X(3341) ∩ X(6)X(2188)
= a^2 (a-b-c) (a^3+a^2 (b-c)-a (b-c)^2-(b-c) (b+c)^2) (a^3-a (b-c)^2+a^2 (-b+c)+(b-c) (b+c)^2) (a^9+3 a^8 (b+c)+4 a^2 b (b-c)^4 c (b+c)-(b-c)^6 (b+c)^3-6 a^5 (b^2-c^2)^2+8 a^3 (b^2-c^2)^2 (b^2+c^2)+a^6 (-8 b^3+4 b^2 c+4 b c^2-8 c^3)+2 a^4 (b-c)^2 (3 b^3-b^2 c-b c^2+3 c^3)-a (b^2-c^2)^2 (3 b^4+10 b^2 c^2+3 c^4)) : : (barys)
= lies on these lines: {3,3341}, {6,2188}, {25,1436}, {56,64}
=ETC 6-9-13 search numbers [1.50993671784466044, 1.82772279812444519, 1.67842405958530454]
Q5 = Q(X(3348)) = ISOGONAL CONJUGATE OF X(14365)
= a^2 (a^4+b^4+2 b^2 c^2-3 c^4-2 a^2 (b^2-c^2)) (a^4-3 b^4+2 b^2 c^2+c^4+2 a^2 (b^2-c^2)) (5 a^12+(b^2-c^2)^6-10 a^10 (b^2+c^2)+36 a^6 (b^2-c^2)^2 (b^2+c^2)+a^8 (-9 b^4+34 b^2 c^2-9 c^4)-a^4 (b^2-c^2)^2 (29 b^4+54 b^2 c^2+29 c^4)+2 a^2 (b^2-c^2)^2 (3 b^6+13 b^4 c^2+13 b^2 c^4+3 c^6)) : : (barys)
= (16 R^2+SB+SC-4 SW) S^4 + (1536 R^6+64 R^4 SB+64 R^4 SC-16 R^2 SB SC-896 R^4 SW-16 R^2 SB SW-16 R^2 SC SW+4 SB SC SW+160 R^2 SW^2-8 SW^3) S^2 -1024 R^6 SB SC+640 R^4 SB SC SW-128 R^2 SB SC SW^2+8 SB SC SW^3 : :
= lies on these lines: {3,2130}, {6,14092}, {25,64}, {56,7037}, {1073,9786}, {1301,1498}, {1436,2155}, {1620,11589}, {2060,14362}, {15394,17928}
= isogonal conjugate of X(14365)
=ETC 6-9-13 search numbers [3.74187532712576789, 3.20124371999504683, -0.302600552147161575]
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(2) locus locus of P such that ABC, A*B*C* are orthologic = {Euler line} U {circumcircle}
*** let W = orthologic center (ABC, A*B*C*)
*** pairs {P=X(i) ∈ Euler Line, W=X(j)} for these {i,j}: {2,69},{3,3},{4,68},{5,3519},{20,4},{21,72},{22,6},{23,895},{24,15316},{25,6391},{26,15317},{30,265},{186,5504},{376,4846},{401,290},{548,14861},{550,3521},{858,67},{1370,66},{1657,21400},{1658,16867},{1817,1439},{2071,74},{2475,18123},{2937,15002},{3146,15077},{3151,8044},{3522,15740},{3534,18550},{3843,14841},{4184,71},{4225,73},{4226,879},{4230,2435},{4236,10099},{5059,15749},{5189,18125},{6636,1176},{7391,18124},{7396,16774},{7471,14220},{7488,54},{7493,5486},{7560,1246},{8613,8795},{8703,13623},{10296,11564},{10298,3431},{10565,17040},{11413,64},{11414,3527},{11634,10097},{12225,6145},{15329,14380},{15331,16665},{15704,17505},{16049,65},{16386,11744},{18859,11559},{19772,2992},{19773,2993},{21312,3426}
*** pairs {P = X(i) ∈ circumcircle, W = X(j)} for these {i,j}: none
*** some points W = W(X(i)) :
W1 = W(X(27)) = X(3)X(307) ∩ X(4)X(916)
= (a+b-c) (a-b+c) (b+c) (a^2-b^2-c^2) (a^3-b^2 c+c^3-a b (b+c)) (a^3+b^3-b c^2-a c (b+c)) : : (barys)
= lies on these lines: {3,307}, {4,916}, {6,226}, {64,516}, {66,674}, {71,440}, {73,6356}, {74,1305}, {272,1175}, {349,2893}, {912,1243}, {1246,15467}, {2218,7083}, {2772,11744}, {6817,8814}, {8804,21091}
= ETC 6-9-13 search numbers [0.928644861130306464, 4.12697450926022525, 0.354922962667158636]
W2 = W(X(28)) = X(3)X(6511) ∩ X(4)X(912)
= a (b+c) (a^2-b^2-c^2) (a^3+a^2 (b-c)+(b-c)^2 (b+c)+a (b^2-2 b c-c^2)) (a^3+a^2 (-b+c)+(b-c)^2 (b+c)+a (-b^2-2 b c+c^2)) : : (barys)
= lies on these lines: {3,6511}, {4,912}, {6,169}, {54,10202}, {64,517}, {65,23604}, {66,518}, {69,20235}, {71,18674}, {72,21015}, {74,13397}, {81,1175}, {520,3657}, {1177,2836}, {1245,2650}, {2771,11744}, {3874,9028}, {3962,10693}, {8673,10099}, {9940,14528}
= ETC 6-9-13 search numbers [-1.01587836573791640, 6.26163956518048247, -0.225449817492301196]
W3 = W(X(29)) = X(4)X(5906) ∩ X(6)X(1210)
= (b+c) (-a^2+b^2+c^2) (a^5-a^2 (b-c)^2 c+a b (b-c)^2 (b+c)+c (b^2-c^2)^2-a^3 (2 b^2-b c+c^2)) (a^5-a^2 b (b-c)^2+a (b-c)^2 c (b+c)+b (b^2-c^2)^2-a^3 (b^2-b c+2 c^2)) : :
= lies on these lines: {4,5906}, {6,1210}, {64,515}, {66,8679}, {73,18641}, {2779,11744}
= ETC 6-9-13 search numbers [5.90756029059245174, 4.51279433805320889, -2.21014404009512785]
Best regards
Ercole Suppa
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