HYACINTHOS 28884

[Kadir Altintas]:

 

Let ABC be a triangle with O, K  the circumcenter and the symmedian point resp.

 

Denote:

 

DEF = the  circumcevian triangle of K.

Ka, Kaa = the  symmedian points of ABD and ADC resp. 

Define Kb, Kbb, Kc, Kcc cyclically.

 

(1) Prove that Ka, Kaa, Kb, Kbb, Kc, Kcc lie on same conic with perspector K.

 

(2) the center of the conic lies on OK line and has simple first barycentrics:

 

X = a^2 (-2 a^2+5 (b^2+c^2)) ::

 

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[Ercole Suppa]

 

 

*** X = X(3)X(6) ∩ X(20)X(14537)  

 

= a^2 (-2 a^2+5 (b^2+c^2)) : : (barys)

 

= 7 S^2-7 SB SC+3 SB SW+3 SC SW : : (barys)

 

= lies on these lines: {3,6}, {20,14537}, {30,9698}, {99,6683}, {115,3628}, {140,7765}, {230,12108}, {232,14865}, {538,7824}, {546,1506}, {548,9300}, {549,7755}, {550,7753}, {620,7852}, {625,7847}, {631,5309}, {632,5254}, {671,16922}, {940,21538}, {1015,3746}, {1194,7496}, {1500,5563}, {1571,7982}, {1656,11648}, {1975,15482}, {2482,7819}, {2548,3529}, {2549,3090}, {3055,12812}, {3091,7603}, {3146,5475}, {3292,8041}, {3520,14581}, {3523,7739}, {3524,5319}, {3525,7738}, {3544,18424}, {3627,3815}, {3767,10303}, {3849,7858}, {3934,7783}, {4045,7874}, {4383,21517}, {5070,18362}, {5215,7827}, {5283,17572}, {5306,15712}, {5346,21843}, {6292,6390}, {6337,7822}, {7618,14001}, {7622,7902}, {7736,17538}, {7745,12103}, {7747,15704}, {7749,14869}, {7757,7780}, {7763,7853}, {7764,7873}, {7769,7861}, {7771,7805}, {7777,7842}, {7781,9466}, {7782,7804}, {7786,7816}, {7791,7821}, {7794,8359}, {7799,7849}, {7801,16043}, {7817,7907}, {7830,7845}, {7831,7895}, {7833,7843}, {7848,7906}, {7854,15810}, {7863,8362}, {7864,7886}, {7876,7880}, {7882,7904}, {7888,11287}, {7891,7915}, {7991,9619}, {8367,15300}, {8716,17130}, {12815,16239}, {14002,15302}, {14901,15034}, {15301,17128}, {16021,16022}, {16589,17531}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,39,5007}, {3,5007,187}, {3,9605,22331}, {3,22331,5206}, {3,22332,7772}, {6,574,15602}, {6,15515,15513}, {32,574,15815}, {32,5024,39}, {32,15815,8589}, {39,187,5041}, {39,5008,9605}, {39,8589,32}, {39,15513,6}, {39,15602,15513}, {549,9607,7755}, {550,9606,7753}, {574,5013,39}, {574,5024,8589}, {5013,15815,5024}, {5024,15815,32}, {5206,9605,5008}, {7764,8356,7873}, {7772,22332,39}, {7781,11285,9466}, {15513,15602,15515}

 

= (6-8-13) search numbers [2.17143868549719377, 2.38618622952132202, 0.986486929701682117]

 

 

Best regards

Ercole Suppa