[Antreas P. Hatzipolakis]:
Let ABC be a triangle, Q a fixed point, QaQbQc the pedal triangle of Q, P a variable point and PaPbPc the pedal triangle of P.
Denote:
P1, P2, P3 = the orthogonal projections of P on QQa, QQb, QQc, resp.
1. Q = H:
The triangles PaPbPc, P1P2P3 are perspective.
Which is the perspector in terms of P?
And which is the locus of the perspector as P moves on a line, the Euler line for example?
2. Q = O
Which is the locus of P such that PaPbPc, P1P2P3 are persepctive?
------------------------------ ------------------------------ ------------------------------
[Ercole Suppa]
(1) Q=H
*** the perspector in terms of P(x,y,z) (barys) is the point: Z=Z(P)=-x (-a^2 x^2+b^2 x^2+c^2 x^2-a^2 x y+b^2 x y-c^2 x y-a^2 x z-b^2 x z+c^2 x z-2 a^2 y z) : : (barys)
*** the locus of the perspector as P moves on the Euler line is the cubic K934
-- K934: ∑ -(b-c)^4 (b+c)^4 (a^2-b^2-c^2)^4 x^3+2 a^2 (a^14-4 a^12 b^2+3 a^10 b^4+8 a^8 b^6-16 a^6 b^8+8 a^4 b^10+3 a^2 b^12-4 b^14+16 a^10 b^2 c^2-22 a^8 b^4 c^2+10 a^6 b^6 c^2+10 a^4 b^8 c^2-22 a^2 b^10 c^2+29 a^6 b^4 c^4-20 a^4 b^6 c^4) x y z+(a-c)^2 (a+c)^2 (a^2-b^2+c^2)^2 (a^8-2 a^6 b^2-a^4 b^4+4 a^2 b^6-2 b^8+6 a^4 b^2 c^2-6 a^2 b^4 c^2+4 b^6 c^2-4 a^4 c^4-2 a^2 b^2 c^4-3 b^4 c^4+4 a^2 c^6+2 b^2 c^6-c^8) y^2 z+(a-b)^2 (a+b)^2 (a^2+b^2-c^2)^2 (a^8-4 a^4 b^4+4 a^2 b^6-b^8-2 a^6 c^2+6 a^4 b^2 c^2-2 a^2 b^4 c^2+2 b^6 c^2-a^4 c^4-6 a^2 b^2 c^4-3 b^4 c^4+4 a^2 c^6+4 b^2 c^6-2 c^8) y z^2 = 0 (barys)
-- K934 passes through ETC points X(i) for these i: 4, 30, 143, 1147, 1992, 2574, 2575, 15460, 15461, 15471, 19136
-- pairs {P=X(i) ∈ Euler line, Z=X(j)} for these {i,j}: {2,1992}, {3,1147}, {4,4}, {5,143}, {25,19136}, {468,15471}, { 1113,15461}, {1114,15460}
-- some points:
Z(X(20)) = X(3)X(15077) ∩ X(4)X(5972)
= (a^2-b^2-c^2) (5 a^4-2 a^2 b^2-3 b^4-2 a^2 c^2+6 b^2 c^2-3 c^4) (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4) : : (barys)
= S^2 (4 R^2-SB-SC-5 SW)+SB SC (-32 R^2+8 SW) : : (barys)
= 4*X[3]-X[15077]
= lies on the cubics K041 and K934 and these lines: {3,15077}, {4,5972},{20,154}, { 69,3522}, {159,11413}, {343, 21734}, {376,5562}, {394,16936}, {511,16879}, {631,11704}, {1092, 8718}, {2071,8907}, {3146,15748}, {3528,12254}, {3619,14118}, { 5059,11064}, {5921,8567}, {6225,16386}, {6467,25406}, {10167, 18732}, {11206,12279}, {12118, 18931}, {19467,22647}
= (6-8-13) search numbers [8.74604547263311570, 7.42474561351121646, -5.53618039173867031]
Z(X(21)) = X(21)X(60) ∩ X(1175)X(18123)
= a^2 (a+b)^2 (a-b-c) (a+c)^2 (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c+a^4 b c+3 a^3 b^2 c-a^2 b^3 c-a b^4 c-a^4 c^2+3 a^3 b c^2+2 a^2 b^2 c^2+a b^3 c^2-b^4 c^2+4 a^3 c^3-a^2 b c^3+a b^2 c^3-a^2 c^4-a b c^4-b^2 c^4-2 a c^5+c^6)0 : : (barys)
= lies on these lines: {21,60}, {1175,18123}
= (6-8-13) search numbers [-1.98600893043271872, 2.66837874465007310, 2.70994485734942944]
Z(X(22)) = X(4)X(15462) ∩ X(22)X(206)
= a^4 (a^2-b^2-c^2) (a^4-b^4-c^4) (a^10-a^8 b^2-2 a^6 b^4+2 a^4 b^6+a^2 b^8-b^10-a^8 c^2-2 a^6 b^2 c^2-2 a^4 b^4 c^2+2 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4-2 a^4 b^2 c^4-2 a^2 b^4 c^4-2 b^6 c^4+2 a^4 c^6+2 a^2 b^2 c^6-2 b^4 c^6+a^2 c^8+3 b^2 c^8-c^10) : : (barys)
= lies on the cubic K934, the curve Q106 and these lines: {4,15462}, {22,206}, {343,19127} , {1176,1899}
= (6-8-13) search numbers [-13.6411454299230452, -14.6085556227188690, 20.0501931883696170]
Z(X(23)) = X(3)X(15077) ∩ X(4)X(5972)
= a^4 (a^4-b^4+b^2 c^2-c^4) (a^6-a^4 b^2-a^2 b^4+b^6-a^4 c^2-a^2 b^2 c^2-2 b^4 c^2-a^2 c^4-2 b^2 c^4+c^6) : : (barys)
= lies on the cubic K934, the curve Q106 and these lines: {4,83}, {23,6593}, {1177,9140}, { 2070,19381}, {3047,12367}, { 5169,19127}, {9979,13315}, {15019,19136}
= (6-8-13) search numbers [-13.7331896236440254, -15.1429069881904160, 20.4626106846442173]
------------------------------ ---------
(2) Q=O
*** the locus of P such that PaPbPc, P1P2P3 are perspective is the cubic K038 (Stammler strophoid).
*** the locus of perspectors W=W(P) is the Euler line of ABC
-- K038: ∑ b^2 (b-c) c^2 (b+c) x^3+4 a^2 (a-b) b^2 (a+b) x y z+a^2 (a-c) (a+c) (2 a^2-2 b^2+c^2) y^2 z-a^2 (a-b) (a+b) (2 a^2+b^2-2 c^2) y z^2 = 0 (barys)
-- K038 passes through ETC points X(i) for these i: 3, 30, 36, 131, 187, 1511, 2482, 3184, 6150, 6592, 12095, 12096, 17729
-- pairs {P=X(i) ∈ K038 , W=X(j)} for these {i,j}: {3,3}, {1511,30}
-- some points:
W(X(36)) = X(2)X(3) ∩ X(35)X(3754)
= a^2 (a^2-b^2+b c-c^2) (a^3-a^2 b-a b^2+b^3-a^2 c-a b c-a c^2+c^3) : : (barys)
= lies on these lines: {2,3}, {35,3754}, {36,214}, {100, 5172}, {191,997}, {515,17009}, { 1125,14794},
{1470,21454}, {1708,4855}, {1737,17010}, {1994, 5398}, {2206,4257}, {2646,8261}, {2771,18861}, {2975,21677},
{3002,5546}, {4256,20966}, {4861,14798}, {5010,5426}, {5204,11684}, {5253,11281}, {5303,18253},
{5445,25440}, {6796,25005}, {10090,11604}, {11263,14792}, {17653,22936}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,4216,6636}, {3,4218,15246}, { 3,19525,17549},
{21,404,442}, { 21,3651,15680}, {404,1006,2}, { 442,5428,21}, {1006,6905,6882}, {1006,21161,5428},
{4188,4189,4190},{4189,15674,21}, {6827,6921,2}, {6830,17566,2}, {11334,19245,13595}
= (6-8-13) search numbers [0.157107355492840441, -0.712827020575484465, 4.06164902438609584]
W(X(131)) = MIDPOINT X(131) AND X(12095)
= 2 a^2 - b^2 - c^2) (5 a^2 - b^2 - c^2) : : (barys)
= 9 S^2 - 9 SB SC - 2 SW^2 : : (barys)
As a point on the Euler line, X() has Shinagawa coefficients {20 R^4 - S^2 - 12 R^2 SW + 2 SW^2, 12 R^4 + S^2 - 4 R^2 SW}
= lies on these lines: {2,3}, {131,12095}, {3564,13557}
= midpoint of X(131) and X(12095)
= (6-8-13) search numbers [-0.116800673126684375, -0.986012472899423574, 4.37719650458860360]
W(X(187)) = W(X(2482)) = MIDPOINT OF X(2) AND X(8598)
= (a^2-b^2-c^2) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (2 a^8-3 a^6 b^2+a^4 b^4-a^2 b^6+b^8-3 a^6 c^2+2 a^4 b^2 c^2+a^2 b^4 c^2-4 b^6 c^2+a^4 c^4+a^2 b^2 c^4+6 b^4 c^4-a^2 c^6-4 b^2 c^6+c^8) : : (barys)
= S^4 + (-20 R^4-SB SC+12 R^2 SW-2 SW^2)S^2 + (-12 R^4+4 R^2 SW)SB SC : : (barys)
= X[115]-3*X[5215], X[625]-2*X[22247], X[6781]+3*X[9167], 3*X[8290]+X[9889], X[8591]+3*X[8859], 2*X[14148]+X[15480]
As a point on the Euler line, X() has Shinagawa coefficients {2 SW^2 - 9 S^2, 9 S^2}
= lies on these lines: {2,3}, {6,7618}, {32,8584}, {69,15655}, {99,9136},{110,6093}, {115,5215}, {141,8588}, {187,524}, {230,543}, {574,597}, {598,11149}, {599,5210}, {620,3849}, {625,22247}, {671,10153}, {1384,1992}, {1499,4786}, {2021,5969}, {2080,5182}, {3053,15534}, {3054,7617}, {3055,7619}, {3564,8593}, {3589,8589}, {3734,5569}, {3815,7622}, {3933,5023}, {5008,20583}, {5032,21309}, {5104,15483}, {5206,7767}, {5305,7782} , {5475,9771}, {5476,9734}, { 5585,21358}, {6781,9167}, {7610,21843}, {7737,11184}, {7750,7870}, {7789,7810}, {7820,15810}, {7891,9939}, {8030,14567}, { 8290,9889}, {8591,8859}, {8860,11164}, {9486,16317}, {9489, 25423}, {9741,22253} ,{11151,11171}, {11161,14830}, {11162,14666}, {11163,12040}, {11645, 19662}, {14148,15480}, {15993, 19911}
= midpoint of X(i) and X(j) for these {i,j}: {2,8598}, {99,22329}, {187,2482}, {376,1513}, {1551,10295}, { 6661,10997}, {7426,7472}, {8352, 9855}
= reflection of X(i) in X(j) for these {i,j}: {381,10011}, {625,22247}, {6390,2482}, {8352,8355}, {22110,620}
= complement of X(8352)
= anticomplement of X(8355)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,376,5077}, {2,8352,8355}, {2,8703,8354}, {2,9855,8352}, {2,11159,3363}, {2,11317,5}, {2,13586,8598}, {3,8369,8359}, {187,6390,3793}, {548,7807,8357}, {548,8360,7833}, {550,16925,8361}, {599,5210,8182}, {1384,11165,1992}, {3734,5569,11168}, {5077,11288,2}, {7807,7833,8360}, {7820,15810,20582}, {7833,8360,8357}, {8352,8598,9855}, {8359,8369,7819}, {8860,11164,11185}, {8860,11185,16509}, {12040,18907,11163}, {16431,16436,11350}
= (6-8-13) search numbers [0.596582692738171558, -0.274511030366413879, 3.55536472166658645]
W(X(3184)) = MIDPOINT OF X(20) AND X(1559)
= (a^2-b^2-c^2) (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4) (2 a^10-a^8 b^2-8 a^6 b^4+10 a^4 b^6-2 a^2 b^8-b^10-a^8 c^2+16 a^6 b^2 c^2-10 a^4 b^4 c^2-8 a^2 b^6 c^2+3 b^8 c^2-8 a^6 c^4-10 a^4 b^2 c^4+20 a^2 b^4 c^4-2 b^6 c^4+10 a^4 c^6-8 a^2 b^2 c^6-2 b^4 c^6-2 a^2 c^8+3 b^2 c^8-c^10) : : (barys)
= S^4 + (160 R^4 - SB SC - 64 R^2 SW + 6 SW^2)S^2 + (-192 R^4 + 80 R^2 SW - 8 SW^2)SB SC : : (barys)
As a point on the Euler line, X() has Shinagawa coefficients {160 R^4+S^2-64 R^2 SW+6 SW^2,-192 R^4-S^2+80 R^2 SW-8 SW^2}
= lies on McCay butterfly Q046 and these lines: {2,3}, {1503,11589}, {3184,12096}, {5894,14379}, {8057,15427}
= midpoint of X(i) and X(j) for these {i,j}: {20,1559}, {3184,12096}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {20,2060,6616}, {376,3079,20}, {550,13155,20}
= (6-8-13) search numbers [7.52369602990113999, 6.63432840690579112, -4.42480719821246397]
W(X(6150)) = W(X(6592)) = COMPLEMENT OF X(24306)
= (3 a^6-7 a^4 b^2+5 a^2 b^4-b^6-7 a^4 c^2-3 a^2 b^2 c^2+b^4 c^2+5 a^2 c^4+b^2 c^4-c^6) (2 a^10-7 a^8 b^2+10 a^6 b^4-8 a^4 b^6+4 a^2 b^8-b^10-7 a^8 c^2+10 a^6 b^2 c^2-a^4 b^4 c^2-5 a^2 b^6 c^2+3 b^8 c^2+10 a^6 c^4-a^4 b^2 c^4+2 a^2 b^4 c^4-2 b^6 c^4-8 a^4 c^6-5 a^2 b^2 c^6-2 b^4 c^6+4 a^2 c^8+3 b^2 c^8-c^10) : : (barys)
= 16 S^4 + (-47 R^4-16 SB SC+44 R^2 SW-12 SW^2)S^2 + (-3 R^4-4 R^2 SW+4 SW^2)SB SC : : (barys)
As a point on the Euler line, X() has Shinagawa coefficients {47 R^4-16 S^2-44 R^2 SW+12 SW^2,3 R^4+16 S^2+4 R^2 SW-4 SW^2}
= lies on McCay butterfly Q046 and these lines : {2,3}, {930,24385}, {6150,6592}
= midpoint of X(i) and X(j) for these {i,j}: {930,24385},{6150,6592}
= complement of X(24306)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,140,15334}, {140,5501,3628}
= (6-8-13) search numbers [6.52180473388831445, 5.63508012542927226, -3.27060855903049298]
Best regards
Ercole Suppa
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