Δευτέρα 28 Οκτωβρίου 2019

HYACINTHOS 28746

[Τran Quang Hung]:
 
 
Dear geometers,
 
Which is the concurrent point X at here https://artofproblemsolving.com/community/c374081h1754574
 
Let $ABC$ be a triangle inscribed in circle $(O)$. $L$ is its symmedian point. $LA$, $LB$, $LC$ meet $(O)$ again at $A'$, $B'$, $C'$. The sides of triangles $ABC$ and $A'B'C'$ meet at $B_c$, $C_b$, $C_a$, $A_c$, $A_b$, $B_a$ as in figure.

1) Prove that symmedian points of the triangles $BB_cA'$, $CC_bA'$, $CC_aB'$, $AA_cB'$, $AA_bC'$, and $BB_aC'$ lie on a conic $\mathcal{C}$.

2) Prove that focus of $\mathcal{C}$ lie on circle diamter $OL$ and they are reflection in $OL$.
 
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[Ercole Suppa]
 
Dear Tran Quang Hung,
 
the point X is:
 
X = ISOGONAL CONJUGATE X(18840)
 
= a^2 (3 a^2 + b^2 + c^2) :: (barys)
= S^2 - SB SC - 2 SB SW - 2 SC SW : : (barys)
 
= X[7896]-2*X[7915]
 
= lies on these lines: {2,7762}, {3,6}, {4,3172}, {5,7735}, {9,5266}, {20,1285}, {25,251}, {30,5286}, {31,218}, {41,16466}, {55,5280}, {56,609}, {69,7819}, {76,11286}, {81,11343}, {83,183}, {99,7894}, {101,1191}, {112,1593}, {115,3843}, {140,7736}, {141,14023}, {159,15257}, {169,1104}, {172,999}, {193,3933}, {194,1003}, {198,16470}, {217,19347}, {220,595}, {230,1656}, {232,3517}, {237,11402}, {248,3527}, {315,7792}, {316,7851}, {378,8778}, {381,3767}, {382,5254}, {384,7754}, {385,7770}, {393,6756}, {405,5276}, {441,11433}, {524,7795}, {550,7738}, {598,15031}, {599,7822}, {754,7784}, {940,16783}, {942,16780}, {966,17698}, {980,21509}, {988,16667}, {995,3207}, {1078,7878}, {1181,8779}, {1184,3291}, {1186,3511}, {1194,9909}, {1249,7487}, {1385,9575}, {1472,1496}, {1482,1572}, {1506,5070}, {1595,3087}, {1597,1968}, {1598,2207}, {1617,4548}, {1627,7484}, {1657,2549}, {1724,19761}, {1743,3965}, {1914,3295}, {1915,8780}, {1971,14530}, {1975,3972}, {1992,3926}, {1995,5354}, {2070,16308}, {2138,17409}, {2229,16396}, {2241,6767}, {2242,7373}, {2300,20818}, {2493,7506}, {2896,7875}, {3051,3167}, {3052,3730}, {3148,9777}, {3224,3499}, {3303,16785}, {3304,16784}, {3314,10583}, {3329,7793}, {3407,12206}, {3509,16787}, {3523,14930}, {3526,3815}, {3528,14482}, {3534,7739}, {3552,7839}, {3567,9475}, {3579,9593}, {3589,7800}, {3618,3785}, {3629,7758}, {3734,7805}, {3744,17742}, {3749,3991}, {3763,7854}, {3788,7838}, {3830,5309}, {3849,7872}, {3851,5475}, {3934,8667}, {4383,5337}, {4386,9709}, {4426,9708}, {5025,20088}, {5032,11165}, {5054,9300}, {5055,7746}, {5073,5355}, {5077,7802}, {5275,11108}, {5277,16408}, {5283,16418}, {5523,12173}, {5710,16788}, {5938,12167}, {6090,9463}, {6144,7820}, {6392,14033}, {6655,7920}, {6656,16989}, {6660,20977}, {6680,7759}, {6792,15000}, {7375,8974}, {7376,13950}, {7388,13763}, {7389,13644}, {7574,16306}, {7581,21736}, {7585,11292}, {7586,11291}, {7750,7803}, {7751,7804}, {7756,15681}, {7761,7829}, {7763,11288}, {7765,17800}, {7768,7846}, {7769,11163}, {7773,7812},{7774,7807}, {7775,7886}, {7779,7881}, {7780,7808}, {7785,7806}, {7788,7832}, {7797,7823}, {7798,7816}, {7801,7890}, {7809,7942}, {7811,7859}, {7817,7825}, {7818,7852}, {7835,7905}, {7836,7837}, {7840,7945}, {7842,7902}, {7843,7844}, {7845,7867}, {7848,7914}, {7850,7944}, {7857,7858}, {7860,7919}, {7864,14712}, {7869,7882}, {7873,7913}, {7874,7903}, {7880,7916}, {7883,7943}, {7884,7911}, {7885,7932}, {7891,13571}, {7896,7915}, {7897,14043}, {7898,7923}, {7899,7926}, {7900,7901}, {7909,7949}, {7912,16984}, {7917,7930}, {7928,9939}, {7929,7948}, {7931,7946}, {8364,14929}, {8550,8721}, {8744,10594}, {8879,15809}, {8882,19173}, {9310,16483}, {9327,16486}, {9490,18899}, {9592,13624}, {9607,15696}, {9609,13564}, {9620,12702}, {9715,22240}, {9969,20993}, {10306,10315}, {10313,11414}, {10314,11484}, {11313,13758}, {11314,13638}, {11321,16998}, {11335,20023}, {11610,11641}, {11648,15684}, {12164,23128}, {12174,13509}, {12188,12829}, {12203,14532}, {12308,14901}, {13735,27523}, {14003,26869}, {14269,14537}, {14581,18535}, {14602,20854}, {14974,21793}, {14996,21516}, {14997,21540}, {15270,19153}, {15589,16045}, {15720,21843}, {16042,21448}, {16060,17379}, {16061,17349}, {16394,26035}, {16589,16857}, {16918,16995}, {17001,17541}, {17002,17686}, {17597,17736}, {18494,27376}, {19118,27369}, {19125,20960}, {19767,21982}, {20897,26864}
 
= isogonal conjugate of X(18840)
= midpoint of X(8396) and X(8416)
= reflection of X(i) in X(j) for these {i,j}: {7784,7834}, {7896,7915}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,7762,7776}, {2,7893,7879}, {3,6,9605}, {3,32,1384}, {3,5093,3095}, {3,9605,5024}, {3,21309,32}, {4,5304,5305}, {6,32,3}, {6,574,22246}, {6,800,15851}, {6,1333,5120}, {6,1384,5024}, {6,2220,4254}, {6,3053,39}, {6,4252,4253}, {6,4258,386}, {6,5008,21309}, {6,5013,7772}, {6,5052,5093}, {6,6423,3312}, {6,6424,3311}, {6,12963,6422}, {6,12968,6421}, {6,13345,8573}, {6,21309,1384}, {6,22331,5013}, {32,39,3053}, {32,187,22331}, {32,5007,6}, {32,5041,5023}, {32,7772,187}, {32,13356,2080}, {32,14075,7772}, {39,187,15515}, {39,3053,3}, {39,5206,15815}, {39,15515,5013}, {41,21764,16466}, {83,6179,183}, {172,5332,16502}, {172,16502,999}, {187,5013,3}, {187,7772,5013}, {187,14075,6}, {193,14001,3933}, {230,2548,1656}, {251,5359,25}, {315,7792,7866}, {316,7856,7851}, {384,7766,7754}, {385,7787,7770}, {574,5023,3}, {609,5299,56}, {1078,7878,11174}, {1384,5024,15655}, {1384,9605,3}, {1692,2031,2080}, {1975,7760,22253}, {2207,10311,1598}, {2242,16781,7373}, {3053,15815,5206}, {3329,7793,11285}, {3618,3785,8362}, {3629,7789,7758}, {3767,7745,381}, {3788,7838,9766}, {3793,8362,3785}, {3972,7760,1975}, {4264,5037,6}, {4383,5337,21526}, {5007,5008,32}, {5007,21309,9605}, {5013,22331,187}, {5032,19661,11165}, {5052,13357,3095}, {5085,5188,3}, {5093,13357,9605}, {5206,15815,3}, {5254,7737,382}, {5280,7031,55}, {5305,18907,4}, {5306,7745,3767}, {5319,7737,5254}, {5368,7747,5309}, {5475,7755,13881}, {5475,13881,3851}, {6421,12968,6398}, {6422,12963,6221}, {6680,7759,7778}, {7750,7803,11287}, {7768,7846,7868}, {7772,15515,39}, {7772,22331,3}, {7773,7828,11318}, {7779,7892,7881}, {7780,7808,15271}, {7785,7806,7887}, {7797,7823,7841}, {7812,7828,7773}, {7822,7826,599}, {7832,7877,7788}, {7854,7889,3763}, {8573,10317,1384}, {8743,10312,25}, {12150,14614,11286}, {16989,20065,6656}
 
= (6-8-13) search numbers [0.657444222365253001, 1.23408524421634083, 2.48286197943525714]
 
 
Best regards
Ercole Suppa
 
στις
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