HYACINTHOS 29288

 

 

 

 Antreas Hatzipolakis

•  

  [APH]:

  
  

  Let ABC be a triangle and A'B'C' the pedal triangle of I.

  
  Denote:
  
  Oa, Ob, Oc = the circumcenters of IBC, ICA, IAB, resp.
  
  La, Lb, Lc = the Euler lines of IBC, ICA, IAB, resp.
  
  Lab, Lac = the parallels through Oa to Lb, Lc, resp.
  Lbc, Lba = the parallels through Ob to Lc, La, resp.
  Lca, Lcb = the parallels through Oc to La, Lb, resp.
  
  A* = Lbc /\ Lcb
  B* = Lca /\ Lac
  C* = Lab /\ Lba
  
  1, A'B'C', A*B*C* are homothetic.
  Homothetic center?
  
  2. ABC, A*B*C* are orthologic.
  Orthologic center (ABC, A*B*C*) = I
  Orthologic center (A*B*C*, ABC) = ?
  
  3. The orthocenter of A*B*C* lies on the Euler line of ABC.
  Point?

  
  
  

  [César Lozada]:

   

  1)      Q = X(1)X(30) ∩ X(21)X(60)

  = a*(-a+b+c)*(a^5-(b+c)*a^4-(2*b^2+3*b*c+2*c^2)*a^3+2*(b+c)*(b^2-3*b*c+c^2)*a^2+(b^2-c^2)^2*a-(b^3+c^3)*(b-c)^2) : : (barys)

  = 2*X(3647)-3*X(16370), 4*X(6701)-3*X(17532), X(16140)-4*X(24929)

  = lies on these lines: {1, 30}, {11, 26725}, {21, 60}, {35, 16139}, {55, 758}, {56, 10122}, {65, 3651}, {78, 18253}, {104, 33667}, {191, 3601}, {200, 21677}, {354, 18444}, {390, 17482}, {442, 1837}, {950, 11263}, {958, 31938}, {997, 15670}, {1479, 33592}, {1697, 16126}, {1749, 7508}, {2475, 3486}, {2771, 10058}, {3057, 3957}, {3065, 5424}, {3255, 6596}, {3576, 5427}, {3584, 12738}, {3612, 5428}, {3647, 16370}, {3652, 5693}, {3671, 10123}, {3746, 14988}, {3962, 4640}, {4305, 18977}, {4313, 14450}, {4666, 11281}, {5426, 13384}, {5698, 15677}, {5919, 10698}, {6701, 17532}, {6841, 11375}, {6938, 16116}, {7680, 17718}, {10572, 14526}, {10738, 33593}, {10950, 31419}, {11604, 12743}, {12635, 20835}, {12688, 21669}, {12740, 14100}, {15726, 16133}, {17606, 31254}, {17728, 18443}

  = midpoint of X(i) and X(j) for these {i,j}: {5441, 16152}, {6938, 16116}

  = reflection of X(i) in X(j) for these (i,j): (1836, 3649), (3652, 6914), (11684, 4640), (17637, 10391)

  = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 16132, 3649), (21, 17637, 16141), (2646, 17637, 21)

  = [ 2.3594864275788710, 2.3577211193343430, 0.9194022788703560 ]

   

  2)      Q* =  X(1)X(30) ∩ X(21)X(104)

  = a*(a^6-(b+c)*(2*a^2-4*b^2+5*b*c-4*c^2)*a^3-(b^2-b*c+c^2)*a^4-(b^4-4*b^2*c^2+c^4)*a^2-(b^2-c^2)*(b-c)*((2*b^2-b*c+2*c^2)*a-b^3-c^3)) : : (barys)

  = 3*X(1)+X(16143), 2*X(5)-3*X(26725), 3*X(21)-2*X(22936), X(191)-3*X(3576), 4*X(1385)-X(3652), 3*X(1385)-X(22936), 3*X(3576)-2*X(5428), 5*X(3616)-2*X(22798), 2*X(3647)-3*X(28443), 2*X(3649)+X(18481), 3*X(3652)-4*X(22936), 3*X(3653)-2*X(15670), X(11684)-4*X(13624), X(11684)-3*X(21161), X(12699)-4*X(16137), X(13465)-3*X(28443), 4*X(13624)-3*X(21161), 3*X(16132)-X(16143), 2*X(18253)-3*X(28465), 3*X(21161)-2*X(22937)

  = lies on these lines: {1, 30}, {3, 758}, {4, 33592}, {5, 26725}, {10, 12738}, {21, 104}, {35, 11571}, {40, 16126}, {100, 13145}, {140, 6326}, {191, 3576}, {355, 442}, {381, 30143}, {515, 11263}, {517, 3651}, {548, 5538}, {944, 2475}, {952, 5499}, {956, 31938}, {960, 13151}, {991, 29097}, {997, 18253}, {999, 10122}, {1006, 5694}, {1319, 17637}, {1482, 16117}, {1483, 11014}, {2646, 13369}, {3579, 4018}, {3616, 22798}, {3647, 13465}, {3653, 15670}, {3654, 3811}, {3754, 18524}, {3916, 4511}, {3957, 11278}, {4653, 5492}, {5426, 7701}, {5690, 11277}, {5731, 14450}, {5787, 5886}, {5882, 12737}, {5885, 6905}, {5901, 16160}, {6001, 24299}, {6175, 28204}, {6264, 13146}, {6598, 6907}, {6675, 8583}, {6701, 18525}, {6853, 9803}, {6912, 31828}, {6914, 15071}, {6924, 15016}, {6940, 22935}, {7489, 31803}, {7986, 19765}, {8261, 10202}, {9856, 15178}, {9943, 33596}, {9956, 31254}, {10052, 16142}, {10074, 33667}, {10246, 12114}, {10572, 13273}, {10902, 14988}, {11012, 24475}, {12104, 19919}, {12608, 33594}, {12709, 24929}, {12740, 30538}, {15680, 16116}, {15911, 28186}, {16113, 30264}, {16120, 30283}, {16133, 30284}, {16141, 21842}, {16146, 18448}, {16147, 18456}, {16151, 18454}, {21677, 26446}, {24680, 33557}, {26202, 28461}, {26285, 31660}, {27086, 32612}, {28174, 31651}

   

  = midpoint of X(i) and X(j) for these {i,j}: {1, 16132}, {40, 16126}, {944, 2475}, {1482, 16117}, {6264, 13146}, {15680, 16116}, {16159, 18481}

  = reflection of X(i) in X(j) for these (i,j): (4, 33592), (21, 1385), (79, 33668), (191, 5428), (355, 442), (3652, 21), (5690, 11277), (6841, 11281), (7701, 31649), (11684, 22937), (13465, 3647), (16138, 13743), (16139, 3), (16159, 3649), (16160, 5901), (19919, 12104), (22937, 13624)

  = X(6288)-of-2nd circumperp triangle

  = X(10610)-of-hexyl triangle

  = X(16132)-of-anti-Aquila triangle

  = X(16139)-of-ABC-X3 reflections triangle

  = X(22804)-of-excentral triangle

  = X(33592)-of-anti-Euler triangle

  = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1385, 21740, 6265), (1385, 26201, 104), (3649, 7354, 79)

   

  = [ 5.2955045433384260, 5.2859939451710390, -2.4629488078286910 ]

   

  3)      X(3651)

   

  César Lozada