HYACINTHOS 29333

[Kadir Altintas]

Let O be the circumcenter of ABC and DEF the  circuncevian triangle of O.
Let Ha be the orthocenter of AFE (=X(4) of AFE) and Ka the symmedian point of AFE (=X(6) of AFE). 

Define Hb,Kb,Hc,Kc cyclically.
Lines HaKa, HbKb, HcKc concur. (i.e. X(4)X(6) lines of AFE, BFD, CDE concur.) 

Concurrency point has first barycentric coordinate:

X=(a^4-(b^2-c^2)^2)^2(a^4-2b^2c^2-a^2(b^2+c^2)::

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[Ercole Suppa]

X = REFLECTION OF X(4) IN X(53)

= (a^2+b^2-c^2)^2 (a^2-b^2+c^2)^2 (a^4-a^2 b^2-a^2 c^2-2 b^2 c^2) : : (barys)

= (4 R^2 SB+4 R^2 SC+SB SC-SB SW-SC SW)S^2 - 4 R^2 SB SC SW+SB SC SW^2 : :

= 2*X[389]-X[6751]

= lies on these lines: {2,26870}, {3,95}, {4,6}, {5,17907}, {22,324}, {24,157}, {25,98}, {54,19212}, {154,436}, {182,458}, {186,2453}, {232,13860}, {275,11402}, {297,1352}, {317,3564}, {340,11898}, {389,6751}, {403,16324}, {427,9744}, {428,14495}, {467,11442}, {511,9308}, {648,1351}, {917,32704}, {930,23233}, {1093,1598}, {1141,23232}, {1300,30247}, {1593,11257}, {1594,23333}, {1597,11169}, {1896,4186}, {1948,24320}, {1993,19174}, {2871,6403}, {2980,10594} ,{3168,17810}, {3172,12110}, {4230,15928}, {4994,11423}, {5020,15466}, {5094,14165}, {5198,14249}, {5422,30506}, {5890,9792}, {6524,6995}, {6525,7714}, {6528,12188}, {6747,11550}, {6750,18381}, {6755,11245}, {6759,8887}, {6820,14826}, {9753,16318}, {9755,10311}, {10519,32000}, {11331,24206}, {11412,19197}, {13200,15356}, {15087,19177}, {16263,17983}, {17984,18027}, {18916,18953}, {19128,19156}, {19544,31623}

= reflection of X(i) in X(j) for these {i,j}: {4,53}, {6751,389}, {18437,5}, {20477,3}

= barycentric product of X(i) and X(j) for these {i,j}: {4,458}, {107,23878}, {182,2052}, {183,393}, {264,10311}, {1096,3403}, {2207,20023}, {3288,6528}

= barycentric quotient of X(i) and X(j) for these {i,j}: {182,394}, {183,3926}, {393,262}, {458,69}, {1096,2186}, {2052,327}, {2207,263}, {3288,520}, {6784,3269}, {10311,3}, {20031,6037}, {23878,3265}, {32713,26714}

= trilinear product of X(i) and X(j) for these {i,j}: {19,458}, {19,458}, {92,10311}, {158,182}, {158,182}, {183,1096}, {183,1096}, {823,3288}, {823,3288}, {2207,3403}, {2207,3403}, {6784,23999}, {6784,23999}, {23878,24019}

= trilinear quotient of X(i) and X(j) for these {i,j}: {158,262}, {183,326}, {393,2186}, {2207,3402}, {3403,3926}, {24019,26714}

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,393,6530}, {4,1249,14853}, {4,14912,3087}, {1629,2052,25}, {3087,15258,14912}, {6530,16264,4}, {8795,8884,19189}, {20477,20792,3}

Best regards,
Ercole Suppa