HYACINTHOS 28625

28625Re: Reflections, Locus

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• Antreas Hatzipolakis

  Nov 12, 2018

   

  [Alexandr Skutin]:

   

  Let ABC be a triangle and P a point on the Euler line.

   

  Denote: 

   

  Ka, Kb, Kc = the symmedian points of PBC, PCA, PAB, resp. 

   

  The reflections of PKa,, PKb, PKc in BC, CA, AB, resp. are concurrent. 

   

  The point of concurrence, as P moves on the Euler line, is a conic. 

   

  *********************

   

  Questions: 

   

  1. Which are the center and the perspector of the conic? 

   

  2. Which is the entire locus of a variable point P such that the reflections of PKa, PKb, PKc in BC, CA, AB, resp. are concurrent? 

  Euler line + which other things?

   

   

  APH

   

  --------------------------------------------------------------------------------------

   

   

  [Ercole Suppa]

   

   

  (1) The locus of the point of conurrence is the conic q2:  

   

  ∑ b^4 (b-c)^4 c^4 (b+c)^4 (a^2-b^2-c^2) x^2+2 a^6 (a-b)^2 b^2 (a+b)^2 (a-c)^2 c^2 (a+c)^2 y z = 0

   

   

  *** center of q2 :

   

  Z =  X(2)X(98) ∩ X(3)X(7731)

   

  = -a^2 (a^10+2 a^6 b^2 c^2-2 a^8 (b^2+c^2)-b^2 c^2 (b^2-c^2)^2 (b^2+c^2)+2 a^4 (b^6+c^6)-a^2 (b^8-b^6 c^2+3 b^4 c^4-b^2 c^6+c^8)) : : (barys)

  = (6 R^4-7 R^2 SB-7 R^2 SC-11 R^2 SW+2 SB SW+2 SC SW+2 SW^2) S^2 + 7 R^2 SB SC SW-2 SB SC SW^2: : (barys)

   

  = lies on these lines: {2,98}, {3,7731}, {54,1511}, {140,11597}, {146,10984}, {399,15056}, {568,12228}, {569,12383}, {1112,15107}, {1176,6593}, {1216,2914}, {1539,8718}, {1614,14643}, {1986,7691}, {2888,10114}, {2931,15043}, {2979,19504}, {5157,11061}, {5422,12310}, {5504,13472}, {5900,6699}, {6030,12824}, {6636,13417}, {7485,17847}, {7488,16223}, {7503,12270}, {7512,11557}, {7514,12281}, {7592,12273}, {10117,15080}, {10203,11702}, {10721,18564}, {11746,12834}, {12121,15033}, {12893,15053}, {13434,17702}, {15051,15463}, {15140,19121}, {19126,25321}

   

  = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3047,5642,110}

   

  = (6-8-13) search numbers [3.36731029076433322, 2.89022993412095697, 0.0855928548555627896]

   

   

  *** the perspector of q2 :

   

  W =  X(110)X(9514) ∩ X(5012)X(5661)

   

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

   

  = lies on these lines: {110,9514},{5012,5661},{5201,23061}

   

  = isogonal conjugate of X(7668)

  = (6-8-13) search numbers [-0.125012972838341376, -0.108892904212601551, 3.77375017228772235]

   

   

  (2)  Locus of points P such that the reflections of PKa, PKb, PKc in BC, CA, AB, resp. are concurrent:  

   

  Γ={GO= Euler line} U {NPC of ABC}

   

   

  Best regards

  Ercole Suppa