[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 DEF is perspective with MaMbMc at
Q = {x (y+z) (2 x^2-(y-z)^2+x (y+z)),y (x+z) (-x^2+2 y^2+y z-z^2+x (y+2 z)),-(x+y) z (x^2+y^2-y z-2 z^2-x (2 y+z))}
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[Ercole Suppa]
Let Q = Q(P) the perspector of DEF and MaMbMc
*** Q = {x (y+z) (2 x^2-(y-z)^2+x (y+z)),y (x+z) (-x^2+2 y^2+y z-z^2+x (y+2 z)),-(x+y) z (x^2+y^2-y z-2 z^2-x (2 y+z))}
*** Pairs {P=X(i),Q=X(j)}: {1, 2650}, {2, 2}, {4, 11381}, {8, 3893}
*** Some point Q = Q(P)
--- Q(X(3)) = X(3)X(54) ∩ X(4)X(11197)
= a^2 (a^2-b^2-c^2)^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (2 a^6-3 a^4 b^2+b^6-3 a^4 c^2-b^4 c^2-b^2 c^4+c^6) : : (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)
= 2*X[4]-3*X[11197], 7*X[3090]-6*X[14635], 5*X[3091]-6*X[10184], 7*X[3523]-6*X[12012]
= lies on these lines: {3,54}, {4,11197}, {5,14918}, {30,14978}, {160,6293}, {185,20775}, {216,14531}, {417,3917}, {418,5562}, {511,26897}, {577,8565}, {852,11793}, {1216,2972}, {3090,14635}, {3091,10184}, {3523,12012}, {3575,26166}, {6368,14329}, {6638,11444}, {6641,17834}, {11413,23709}, {15905,26216}, {18564,25043}
= barycentric product of X(i) and X(j) for these {i,j}: {343, 13367}, {394, 3574}, {418, 26166}, {5562, 23292}
= trilinear product of X(i) and X(j) for these {i,j}: {255, 3574}, {418, 17859}, {418, 17859}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,11412,13409}, {97,7691,3}
= (6-8-13) search numbers [79.7662647476283731, -21.4493895352647752, -18.3249572618146410]
--- Q(X(5)) = X(2)X(22269) ∩ X(5)X(23607)
= (a^2 b^2-b^4+a^2 c^2+2 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^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-4 a^4 b^2 c^2-3 a^2 b^4 c^2+6 b^6 c^2-3 a^4 c^4-3 a^2 b^2 c^4-10 b^4 c^4+3 a^2 c^6+6 b^2 c^6-c^8) : : (barys)
= 3 S^4 + (16 R^2 SB+16 R^2 SC+9 SB SC-12 R^2 SW-4 SB SW-4 SC SW+3 SW^2)S^2 -16 R^4 SB SC+SB SC SW^2 : : (barys)
= 6*X[2]-5*X[22269], 7*X[3090]-6*X[12013]
= lies on these lines: {2,22269}, {5,23607}, {381,2888}, {1209,11197}, {3078,14978}, {3090,12013}
= (6-8-13) search numbers [-55.9113696154374546, 32.2119327499516240, 7.14534316983515691]
--- Q(X(6)) = REFLECTION OF X(23642) IN X(6)
= a^2 (b^2+c^2) (2 a^4+a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) : : (barys)
= (8 R^2 SB+8 R^2 SC-4 R^2 SW-3 SB SW-3 SC SW+2 SW^2) S^2 -2 SB SC SW^2-SB SW^3-SC SW^3 : : (barys)
= lies on these lines: {6,22}, {32,23635}, {69,8878}, {141,23297}, {193,732}, {217,5052}, {511,8152}, {688,22260}, {1843,3051}, {3124,9969}, {3172,12167}, {3231,9822}, {3313,8041}, {3618,13410}, {9971,16285}, {11574,20965}
= reflection of X(23642) in X(6)
= barycentric product of X(i) and X(j) for these {i,j}: {31, 23665}, {39, 7745}, {427, 21637}, {1843, 6676}, {2084, 18063}
= trilinear product of X(i) and X(j) for these {i,j}: {32, 23665}, {32, 23665}, {688, 18063}, {688, 18063}, {1964, 7745}, {1964, 7745}, {17442, 21637}, {17442, 21637}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {6,23642,11205}
= (6-8-13) search numbers [0.159295888639261934, 0.612955510953873506, 3.14278948725973040]
--- Q(X(7)) = MIDPOINT OF X(20059) AND X(25722)
= a (a^3 b-3 a^2 b^2+3 a b^3-b^4+a^3 c+6 a^2 b c-3 a b^2 c-4 b^3 c-3 a^2 c^2-3 a b c^2+10 b^2 c^2+3 a c^3-4 b c^3-c^4) : : (barys)
= 4*X[7]-3*X[354], 2*X[144]-3*X[210], 2*X[960]-3*X[10861], 4*X[15008]-5*X[18398]
= lies on these lines: {4,10307}, {7,354}, {9,1155}, {37,3000}, {46,5779}, {55,2951}, {56,11372}, {57,3062}, {65,971}, {142,17605}, {144,210}, {226,5918}, {279,10939}, {513,2262}, {516,3057}, {518,1278}, {527,3059}, {960,10861}, {962,9850}, {990,1456}, {1001,8544}, {1122,4014}, {1418,2310}, {1420,24644}, {1445,16112}, {1721,6180}, {1770,5762}, {1864,11246}, {2646,5732}, {2801,17636}, {3304,10384}, {3600,10866}, {3698,5229}, {3748,4326}, {3983,5128}, {4292,12688}, {4295,12680}, {4298,9848}, {4321,20323}, {4336,6610}, {4887,14523}, {4907,7271}, {5221,10398}, {5226,10178}, {5728,17637}, {5784,17768}, {5805,7702}, {5817,24914}, {5843,14872}, {7988,11575}, {8545,11495}, {10442,21334}, {11375,21151}, {12764,18482}, {15008,18398}, {15185,17660}
= midpoint of X(20059 ) and X(25722)
= reflection of X(i) in X(j) for these {i,j}: {65,4312}, {144,15587}, {3057,8581}, {3059,17668}, {14100,7}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {7,14100,354}, {144,15587,210}, {4014,12723,1122}, {7354,17634,3057}, {8545,11495,15837}
= (6-8-13) search numbers [-0.933699416394155389, -0.937515626542905343, 4.72065195400369298]
Best regards
Ercole Suppa
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