Let ABC be a triangle with centroid G and orthocenter H.
The line parallel to BC for the point Pa, such that APa:PaH= t, intersects the sides AC and AB respectively at Ab and Ac.
Let Da be the point of intersection of the lines BAb and CAc.
Let La be the line joining the orthocenters of the triangles BDaAc and CDaAb.
Lb and Lb are defined similarly.
Let G' be the centroid of A'B'C', the triangle bound by La,Lb,Lc.
Let F be the fixed point (not on the line at infinity) of the affine transformation that applies ABC in A'B'C'.
Then, as t varies, the line G'F passes through a fixed point Q:
= X(2)X(1495)∩X(3)X(16655)
= 3a^6-a^4(b^2+c^2)-a^2(b^2-c^2)^2-(b^2-c^2)^2(b^2+c^2) : : .
= lies on these lines: {2,1495}, {3,16655}, {4,54}, {5,3796}, {6,428}, {20,3917}, {22,1352}, {23,11442}, {24,14216}, {25,1503}, {30,394}, {51,6776}, {66,20987}, {68,7517}, {69,16276}, {110,7391}, {125,6353}, {154,427}, {155,7553}, {182,6997}, {185,7487}, {193,21969}, {343,9909}, {373,7398}, {378,16658}, {382,3167}, {393,8779}, {462,5868}, {463,5869}, {468,1853}, {511,7500}, {542,6515}, {1181,6756}, {1204,12324}, {1370,9306}, {1498,3575}, {1501,3767}, {1514,3830}, {1593,16621}, {1595,19357}, {1596,18396}, {1597,16654}, {1598,6146}, {1843,5596}, {1885,15811}, {1907,11425}, {1971,22075}, {1994,20423}, {2165,2980}, {2182,5101}, {2883,12173}, {2979,20062}, {3051,3331}, {3060,7519}, {3088,13367}, {3091,11572}, {3146,3292}, {3147,20299}, {3148,8721}, {3426,3534}, {3515,6247}, {3518,11457}, {3541,10282}, {3542,18381}, {3867,19125}, {4232,23291}, {4549,18435}, {5012,7394}, {5064,23292}, {5094,10192}, {5133,6800}, {5198,12241}, {5200,5871}, {5310,12588}, {5320,5800}, {5322,12589}, {5422,11179}, {5480,11402}, {5651,7386}, {5654,10540}, {5878,6240}, {6000,18533}, {6053,10706}, {6243,9936}, {6623,13851}, {6696,15750}, {7387,12134}, {7392,25406}, {7401,10984}, {7408,13366}, {7493,21243}, {7499,10516}, {7507,16252}, {7540,18445}, {7576,11456}, {7667,17811}, {7714,11433}, {7715,18914}, {7716,26926}, {7795,10328}, {8550,9777}, {8780,11064}, {9707,15559}, {9714,12359}, {9934,12140}, {10151,18405}, {10301,11245}, {10535,11393}, {10539,14790}, {11392,26888}, {11403,16656}, {12112,18559}, {12174,13568}, {12290,20427}, {13399,18931}, {13417,14683}, {13595,18911}, {14227,19219}, {14912,15004}, {15152,18386}, {15448,23332}, {15583,19118}, {16105,23236}, {16319,18870}, {16657,18535}, {17907,19558}, {18374,23327}, {18378,25738}, {22135,27376}.
= reflection of X(i) in X(j), for these {i, j}: {1370,9306}, {1899,25}, {18396,1596}.
(6 - 9 - 13) - search numbers of Q: (-5.64655890503856,-7.54773956309683,11.4721267509923).
Angel Montesdeoca
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