Let ABC be a triangle and P a point.
Denote:
A'B'C' = the antipedal triangle of P
A"B"C" = the 1. pedal or 2. cevian triangle of the isogonal conjugate P
MaMbMc = the medial triangle of A"B"C"
Which are the loci 1, 2 such that A'B'C', MaMbMc are perspective?
O lies on the loci (Thanassis Gakopoulos, Romantics of Geometry 3441)
[Ercole Suppa]:
Let P(x:y:z) be the barycentric coordinates of P and let Q = Q(P) be the point of concurrency.
1. Locus: the entire plane
*** Pairs {P = X(i),Q(P) = X(j)} with 1<=i<=150, for these {i,j} :
{1,1}, {2,14482}, {3,6}, {4,2}, {6,5544}, {20,4}, {40,57}, {54,5643}, {64,3}, {74,110}, {84,9}, {98,99}, {99,98}, {100,104}, {101,103}, {102,109}, {103,101}, {104,100}, {105,1292}, {106,1293}, {107,1294}, {108,1295}, {109,102}, {110,74}, {111,1296}, {112,1297}
*** Some points:
Q(X(5)) = X(39)X(16040) ∩ X(231)X(3054)
= 4 a^12-8 a^10 b^2-5 a^8 b^4+21 a^6 b^6-13 a^4 b^8-a^2 b^10+2 b^12-8 a^10 c^2-2 a^8 b^2 c^2+15 a^6 b^4 c^2+3 a^4 b^6 c^2-3 a^2 b^8 c^2-5 b^10 c^2-5 a^8 c^4+15 a^6 b^2 c^4+20 a^4 b^4 c^4+4 a^2 b^6 c^4+2 b^8 c^4+21 a^6 c^6+3 a^4 b^2 c^6+4 a^2 b^4 c^6+2 b^6 c^6-13 a^4 c^8-3 a^2 b^2 c^8+2 b^4 c^8-a^2 c^10-5 b^2 c^10+2 c^12 : : (barys)
= (144 R^4-30 R^2 SB-30 R^2 SC-5 SB SC-96 R^2 SW+12 SB SW+12 SC SW+15 SW^2)S^2 + 3 S^4+4 R^2 SB SC SW-SB SC SW^2 : : (barys)
= lies on these lines: {39,16040}, {231,3054}, {550,10610}
= (6-9-13) search numbers: [-37.4604759119342082378, -2.9105597347857869649, 22.9451177961134046501]
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Q(X(7)) = X(2)X(955) ∩ X(4)X(3601)
= 3 a^6-8 a^5 b+a^4 b^2+12 a^3 b^3-7 a^2 b^4-4 a b^5+3 b^6-8 a^5 c+6 a^4 b c+4 a^3 b^2 c+4 a b^4 c-6 b^5 c+a^4 c^2+4 a^3 b c^2+14 a^2 b^2 c^2-3 b^4 c^2+12 a^3 c^3+12 b^3 c^3-7 a^2 c^4+4 a b c^4-3 b^2 c^4-4 a c^5-6 b c^5+3 c^6 : : (barys)
= lies on these lines: {2,955}, {4,3601}, {11,3488}, {65,3085}, {72,18231}, {100,954}, {165,226}, {405,5328}, {442,5703}, {498,18397}, {950,7988}, {2550,3158}, {3475,5659}, {3651,5217}, {5226,7580}, {5308,15252}, {5316,16845}, {5660,5817}, {5748,13615}, {6843,24929}, {6908,11374}, {8164,18446}, {10393,10588}, {10578,33108}, {11041,31397}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {226,5218,5759}
= (6-9-13) search numbers: [1.8805725965928889516, 1.6721640452867109164, 1.6150558675122458757]
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Q(X(8)) = X(2)X(12915) ∩ X(57)X(8166)
= 3 a^6-4 a^5 b-7 a^4 b^2+12 a^3 b^3+a^2 b^4-8 a b^5+3 b^6-4 a^5 c+22 a^4 b c-12 a^3 b^2 c-32 a^2 b^3 c+32 a b^4 c-6 b^5 c-7 a^4 c^2-12 a^3 b c^2+62 a^2 b^2 c^2-24 a b^3 c^2-3 b^4 c^2+12 a^3 c^3-32 a^2 b c^3-24 a b^2 c^3+12 b^3 c^3+a^2 c^4+32 a b c^4-3 b^2 c^4-8 a c^5-6 b c^5+3 c^6 : : (barys)
= lies on these lines: {2,12915}, {57,8166}, {165,497}, {392,18231}, {944,1210}, {956,5704}, {1317,18391}, {1864,5658}, {3057,3086}, {3158,11019}, {3651,5204}, {4000,7658}, {5083,5660}, {5274,17613}, {5603,12736}, {8732,11575}, {14986,20789}
= (6-9-13) search numbers: [1.3583531469500148848, 1.7514781533026336868, 1.8011704617979335270]
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Q(X(9)) = X(57)X(5658) ∩ X(165)X(15299)
= a (a^8-2 a^7 b-2 a^6 b^2+6 a^5 b^3-6 a^3 b^5+2 a^2 b^6+2 a b^7-b^8-2 a^7 c-4 a^6 b c+18 a^5 b^2 c-8 a^4 b^3 c-14 a^3 b^4 c+12 a^2 b^5 c-2 a b^6 c-2 a^6 c^2+18 a^5 b c^2-48 a^4 b^2 c^2+20 a^3 b^3 c^2+46 a^2 b^4 c^2-38 a b^5 c^2+4 b^6 c^2+6 a^5 c^3-8 a^4 b c^3+20 a^3 b^2 c^3-120 a^2 b^3 c^3+38 a b^4 c^3-14 a^3 b c^4+46 a^2 b^2 c^4+38 a b^3 c^4-6 b^4 c^4-6 a^3 c^5+12 a^2 b c^5-38 a b^2 c^5+2 a^2 c^6-2 a b c^6+4 b^2 c^6+2 a c^7-c^8) : : (barys)
= lies on these lines: {57,5658}, {165,15299}, {1058,1210}, {3086,7160}, {3651,10396}, {5219,6983}, {5704,18231}, {7966,18391}, {15285,16610}
= (6-9-13) search numbers: [1.4035175341425998625, 1.7411909844776710268, 1.7874472461263299942]
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Q(X(10)) = X(376)X(3017) ∩ X(551)X(4906)
= 4 a^6-a^4 b^2+9 a^3 b^3+3 a^2 b^4-a b^5+2 b^6+6 a^4 b c+3 a^3 b^2 c+3 a^2 b^3 c+5 a b^4 c-b^5 c-a^4 c^2+3 a^3 b c^2+4 a^2 b^2 c^2-4 a b^3 c^2-2 b^4 c^2+9 a^3 c^3+3 a^2 b c^3-4 a b^2 c^3+2 b^3 c^3+3 a^2 c^4+5 a b c^4-2 b^2 c^4-a c^5-b c^5+2 c^6 : : (barys)
= lies on these lines: {376,3017}, {551,4906}, {3752,14838}, {14829,31168}
= (6-9-13) search numbers: [1.3216345778209715895, 1.8417306536784425427, 1.7556349934434704553]
2. Locus: {Linf} ∪ {circumcircle} ∪ {q9: circum-curve of order 9 passing through X(3)}
q9: ∑ b^4 c^6 (a^2+b^2-c^2) (a^2-b^2+c^2) (2 a^2 b^2-2 b^4-a^2 c^2+3 b^2 c^2+c^4) x^6 y^2 z+2 a^2 b^4 c^6 (a^6+a^4 b^2-5 a^2 b^4+3 b^6+2 a^4 c^2+16 a^2 b^2 c^2-18 b^4 c^2-7 a^2 c^4+11 b^2 c^4+4 c^6) x^5 y^3 z+2 a^2 (a-b) b^2 (a+b) c^8 (3 a^4+26 a^2 b^2+3 b^4-6 a^2 c^2-6 b^2 c^2+3 c^4) x^4 y^4 z-2 a^4 b^2 c^6 (3 a^6-5 a^4 b^2+a^2 b^4+b^6-18 a^4 c^2+16 a^2 b^2 c^2+2 b^4 c^2+11 a^2 c^4-7 b^2 c^4+4 c^6) x^3 y^5 z-a^4 c^6 (a^2-b^2-c^2) (a^2+b^2-c^2) (2 a^4-2 a^2 b^2-3 a^2 c^2+b^2 c^2-c^4) x^2 y^6 z+2 a^2 b^4 (b-c) c^4 (b+c) (a^2-b^2-c^2) (a^4+a^2 b^2-2 b^4+a^2 c^2+14 b^2 c^2-2 c^4) x^5 y^2 z^2+a^2 b^4 c^4 (a^8+a^6 b^2-5 a^4 b^4+3 a^2 b^6+9 a^6 c^2+88 a^4 b^2 c^2-91 a^2 b^4 c^2-6 b^6 c^2-55 a^4 c^4+21 a^2 b^2 c^4+14 b^4 c^4+43 a^2 c^6-10 b^2 c^6+2 c^8) x^4 y^3 z^2-a^4 b^2 c^4 (3 a^6 b^2-5 a^4 b^4+a^2 b^6+b^8-6 a^6 c^2-91 a^4 b^2 c^2+88 a^2 b^4 c^2+9 b^6 c^2+14 a^4 c^4+21 a^2 b^2 c^4-55 b^4 c^4-10 a^2 c^6+43 b^2 c^6+2 c^8) x^3 y^4 z^2+64 a^6 (a-b) b^6 (a+b) c^4 x^3 y^3 z^3-2 a^10 c^4 (a^2-b^2-c^2) (a^2+b^2-c^2) y^6 z^3+2 a^10 b^2 c^2 (a^4-2 a^2 b^2+b^4+4 a^2 c^2+4 b^2 c^2-5 c^4) y^5 z^4-2 a^10 b^2 c^2 (a^4+4 a^2 b^2-5 b^4-2 a^2 c^2+4 b^2 c^2+c^4) y^4 z^5+2 a^10 b^4 (a^2-b^2-c^2) (a^2-b^2+c^2) y^3 z^6 = 0 (barys)
*** Pairs {P = X(i),Q(P) = X(j)} for these {i,j}: {3,6}
Best regards,
Ercole Suppa
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