[Kadir Altintas]:
Let ABC be a triangle, P = (x:y:z) a point and DEF the cevian triangle of P. .
Denote:
D', E', F' = the reflections of D,E,F in P, resp.
MaMbMc = the medial triangle of D'E'F'
Prove that ABC is perspective with MAMbMc at
Q = x (y+z) (9 x^2-3 y^2+10 y z-3 z^2+6 x (y+z)) : :
P=X(1) ---> X=?
P=X(2) ---> X=X(2)
P=X(3) ---> X=?
P=X(4) ---> X=X(22334)
P=X(5) ---> X=?
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[Ercole Suppa]
Let Q = Q(P) the perspector of ABC and MAMbMc
*** Q = x (y+z) (9 x^2-3 y^2+10 y z-3 z^2+6 x (y+z)) : : (barys)
*** Pairs {P=X(i),Q=X(j)}: {2, 2}, {4, 22334}
*** Some point Q = Q(P)
--- Q(X(1)) = X(1)X(3052) ∩ X(10)X(4035)
= a (3 a+3 b-c) (b+c) (3 a-b+3 c),b (3 a+3 b-c) (a+c) (-a+3 b+3 c): : (barys)
= S^4 + (16 R^4+8 R^2 SB+8 R^2 SC-SB SC-16 R^2 SW-2 SB SW-2 SC SW+3 SW^2)S^2-32 R^4 SB SC+20 R^2 SB SC SW-3 SB SC SW^2: : (barys)
= lies on these lines: {1,3052}, {10,4035}, {19,1100}, {37,2650}, {42,3922}, {75,145}, {225,3649}, {267,5425}, {518,17038}, {596,3635}, {759,4658}, {897,17016}, {942,994}, {3057,13476}, {3924,16666}, {3931,4757}, {4004,4646}, {4802,21105}, {4864,23051}, {6737,17392}, {11518,15509}
= barycentric product of X(i) and X(j) for these {i,j}: {1577, 28162}
= trilinear product of X(i) and X(j) for these {i,j}: {523, 28162}, {523, 28162}
= (6-8-13) search numbers [1.46368658137484790, 1.58011619580239262, 1.87119023187125442]
--- Q(X(3)) = ISOGONAL CONJUGATE OF X(19169)
= a^2 (a^2-b^2-c^2) (3 a^4-6 a^2 b^2+3 b^4-2 a^2 c^2-2 b^2 c^2-c^4) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (3 a^4-2 a^2 b^2-b^4-6 a^2 c^2-2 b^2 c^2+3 c^4) : : (barys)
= 3 S^4 + (32 R^4+12 R^2 SB+12 R^2 SC-3 SB SC-32 R^2 SW-3 SB SW-3 SC SW+6 SW^2)S^2 -64 R^4 SB SC+40 R^2 SB SC SW-6 SB SC SW^2 : : (barys)
= 4*X[5]-3*X[8799], 5*X[631]-3*X[13599]
= lies on these lines: {3,13382}, {5,8799}, {20,264}, {216,14531}, {418,8798}, {511,17039}, {548,6662}, {631,13599}, {3528,15318}
= isogonal conjugate of X(19169)
= barycentric product of X(i) and X(j) for these {i,j}: {343, 14528}
= (6-8-13) search numbers [4.66844126591328544, -1.03668018192962617, 2.20370094666798856]
--- Q(X(5)) = X(4)X(95) ∩ X(1656)X(22268)
= (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4-4 a^2 b^2+3 b^4-2 a^2 c^2-4 b^2 c^2+c^4) (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4-4 a^2 c^2-4 b^2 c^2+3 c^4) : : (barys)
= 6 S^4 + (12 R^2 SB+12 R^2 SC+10 SB SC-12 R^2 SW-3 SB SW-3 SC SW+3 SW^2)S^2 -16 R^4 SB SC+SB SC SW^2 : : (barys)
= 5*X[1656]-3*X[22268]
= lies on these lines: {4,95}, {1656,22268}, {3519,3527}, {3854,11282}, {5056,6750}, {5068,15319}
= barycentric product of X(i) and X(j) for these {i,j}: {233, 8797}
= (6-8-13) search numbers [-33.1190762902621515, 18.2303922941479111, 6.30535118146451062]
--- Q(X(6)) = X(6)X(9909) ∩ X(76)X(193)
= a^2 (3 a^2+3 b^2-c^2) (b^2+c^2) (3 a^2-b^2+3 c^2) : : (barys)
= (16 R^2 SB+16 R^2 SC-8 R^2 SW-6 SB SW-6 SC SW+3 SW^2)S^2 -3 SB SC SW^2-3 SB SW^3-3 SC SW^3 : : (barys)
= lies on these lines: {6,9909}, {76,193}, {141,3787}, {511,17042}, {2353,5007}, {6467,27375}
= barycentric product of X(i) and X(j) for these {i,j}: {39, 5395}
= trilinear product of X(i) and X(j) for these {i,j}: {1964, 5395}, {1964, 5395}
= (6-8-13) search numbers [0.652669621413074348, 1.07338485939586171, 2.59631975398120976]
--- Q(X(7)) = X(4)X(5586) ∩ X(8)X(4312)
= a (a^2-2 a b+b^2+6 a c+6 b c-7 c^2) (a^2+6 a b-7 b^2-2 a c+6 b c+c^2) : : (barys)
= lies on the Feuerbach hyperbola and these lines: {4,5586}, {8,4312}, {9,8169}, {516,7320}, {1000,28194}, {1699,10307}, {2346,2951}, {3927,4866}, {7285,15299}, {10390,15726}
= isogonal conjugate of Q*(X(7))
= (6-8-13) search numbers [-0.0264236097857761254, -0.0289203288206526097, 3.67288176022288143]
--- Q*(X(7)) = ISOGONAL CONJUGATE OF Q(X(7))
= a (7 a^2-6 a b-b^2-6 a c+2 b c-c^2) : : (barys)
= lies on these lines: {1,3}, {9,4421}, {10,11106}, {30,5726}, {100,3305}, {200,3219}, {390,10164}, {516,5226}, {902,2999}, {910,3731}, {950,9588}, {993,4915}, {1190,5526}, {1200,3730}, {1323,3599}, {1479,17559}, {1698,4294}, {1699,5218}, {2177,9340}, {2951,7676}, {2975,11519}, {3052,16469}, {3062,15837}, {3158,4640}, {3474,3982}, {3475,4114}, {3523,12575}, {3583,6939}, {3586,19875}, {3624,10624}, {3632,4305}, {3679,4304}, {3689,3929}, {3752,16487}, {3895,17549}, {3911,10385}, {4189,4853}, {4312,9778}, {4315,10304}, {4428,5437}, {4882,8715}, {4995,5219}, {5231,20075}, {5234,5687}, {5267,12629}, {5312,10460}, {5432,7988}, {5493,5703}, {5974,11995}, {6174,20196}, {6284,7989}, {6736,17576}, {7290,21000}, {7322,15430}, {7951,7965}, {7967,8275}, {8164,28150}, {8616,23511}, {8666,12127}, {8833,9582}, {8917,10482}, {9578,15338}, {9589,13411}, {10391,15104}, {10591,19872}, {10860,15298}, {16140,16143}, {18524,18529}, {19541,24644}
= isogonal conjugate of Q(X(7))
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,13370,7280}, {40,24929,18421}, {55,165,1}, {55,1155,10389}, {100,4512,8580}, {165,10980,1155}, {1155,10389,10980}, {1697,5217,7987}, {1697,7987,1}, {2646,11531,1}, {3158,4640,5223}, {3295,3361,1}, {3576,9819,1}, {3601,7991,1}, {3746,15803,1}, {5432,9580,7988}, {9778,13405,4312}, {10389,10980,1}, {11224,13384,1}, {18421,24929,1}
= (6-8-13) search numbers [3.34963376721322456, 3.06045675133870158, -0.0240980846562102486]
--- Q(X(8)) = X(1)X(3848) ∩ X(4)X(3625)
= a (a+b-7 c) (a-b-c) (a-7 b+c) : : (barys)
= 5*X[3617]-3*X[7320], 14*X[14150]-15*X[16853]
= lies on these lines: {1,3848}, {4,3625}, {7,3621}, {9,3893}, {21,2136}, {79,3632}, {80,4816}, {84,12702}, {519,3296}, {1000,3626}, {1392,4420}, {1706,15179}, {3577,11278}, {3617,7320}, {3679,13606}, {3880,4866}, {4677,5560}, {4778,23836}, {4900,15829}, {5221,7091}, {5558,18221}, {6762,7284}, {7982,16615}, {11256,15015}, {12629,16417}, {13602,17575}, {14150,16853}
= (6-8-13) search numbers [5.26783938082415552, 1.49654527864322600, 0.173284190158689326]
Best regards
Ercole Suppa
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