Let A'B'C' be the cevian triangle and A"B"C" the circumcevian triangle of the incenter I of triangle ABC.
The four points of intersection of the circumcircles of BA'I, CA'I with the side lines A"C", A"B", respectively, lie in a circle (Oa), and define (Ob), (Oc) cyclically.
**** The triangles A"B"C" and OaObOc are perpective and parallelogic, with perspector the circumcenter, and parallelogic centers the incenter and
W = MIDPOINT OF X(1) AND X(5400)
= a(a^5(b+c)-4a^4b c+a^3(-2b^3+3b^2c+3b c^2-2c^3)+5a^2b c(b-c)^2+a(b-c)^2(b^3-2b^2c-2b c^2+c^3)- b c(b^2-c^2)^2) : :
= lies on these lines: {1,5}, {3,23404}, {40,1054}, {43,16200}, {56,1777}, {57,1361}, {102,105}, {104,106}, {117,614}, {244,2800}, {386,10595}, {392,25939}, {484,23153}, {515,1149}, {517,1739}, {551,1064}, {581,3622}, {759,953}, {899,28234}, {912,4694}, {946,1201}, {962,28370}, {978,7982}, {995,4000}, {999,6180}, {1086,1537}, {1125,26095}, {1193,13464}, {1318,14511}, {1331,13279}, {1482,3216}, {1616,3149}, {1647,10265}, {1724,10680}, {1742,3576}, {1772,12758}, {2718,28219}, {3073,5563}, {3293,10222}, {3953,5887}, {3976,5693}, {3987,23340}, {4202,19861}, {4657,17044}, {5844,31855}, {6261,28011}, {6684,28352}, {6788,12247}, {8583,25882}, {8686,28233}, {10090,23703}, {11362,27627}, {12245,17749}, {12608,23675}, {12616,28018}, {14217,24715}, {15558,24028}, {16483,22753}, {17154,30196}
= midpoint of X(i) and X(j), for these {i, j}: {1, 5400}, {17154, 30196}
(6 - 9 - 13) - search numbers of W: (6.57398197129883, 10.3135957057535, -6.53366268575215).
**** ABC and the triangle formed from polars of the vertices of ABC, with respect to the circles (Oa), (Ob) and (Oc), are perspective with perspector
P = X(2292)X(3754)∩X(4535)X(18697)
= a(b+c)^2/(2a^2-(b+c)^2) : :
= lies on these lines: {2292,3754}, {4535,18697}
(6 - 9 - 13) - search numbers of P: (-22.1532113815461, -34.2091805639780, 37.5485024330674).
Best regards,
Angel Montesdeoca
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