HYACINTHOS 27208

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  [Antreas P. Hatzipolakis]:

  
  So we have
  
  Let ABC be a triangle, AiBiCi the pedal triangle of I and L a line passing through I
  
  Denote:
  
  A', B', C'  = the orthogonal projections of A, B, C on L, resp.
  
  The circles with diameters A'Ai, B'Bi, C'Ci and the circumcircle of AiBiCi (= incircle) are concurrent

  
  
  [César Lozada]:

  
  
  

  Lines through X(1) are the tripolars of points on the unnamed circum-conic q0 with trilinear equation v*w+w*u+u*v=0, through ETC’s 88, 100, 162, 190, 651, 653, 655, 658, 660, 662, 673, 771, 799, 823, 897, 1156, 1492, 1821, 2349, 2580, 2581, 3257, 4598, 4599, 4604, 4606, 4607, 8052 and vertices of triangles: CONWAY INNER, HONSBERGER.. This conic has center X(9) and perspector X(1).

   

  Suppose P=u:v:w (trilinears) is a point on q0 and L is the tripolar of P. Then the given circles concur at:

  Q = u^4*(-a+b+c)*((a+b-c)*((b^2+2* b*c-c^2)*a-(b-c)*(3*b^2-6*b*c+ c^2))*(a-b+c)^2*u^2*v^2-(a-b+ c)*((b^2-2*b*c-c^2)*a-(b-c)*( b^2-6*b*c+3*c^2))*(a+b-c)^2*u^ 2*w^2+2*c*(b-c)*(a-b+c)*(a+b- c)^3*u*w^3-4*(a+b-c)*(a-b+c)*( b*c*a^2-(b^2-c^2)*(b-c)*a+(b^ 2-3*b*c+c^2)*(b-c)^2)*w*u^3-4* (a+b-c)*(a-b+c)*(b*c*a^2-(b^2- c^2)*(b-c)*a+(b^2-3*b*c+c^2)*( b-c)^2)*u^3*v-2*b*(b-c)*(a+b- c)*(a-b+c)^3*u*v^3-b^2*(a-b+c) ^4*v^4-c^2*(a+b-c)^4*w^4)/a : :

   

  ETC-pairs (P,Q): (88,3326), (651,14027), (653,3326), (658,3323)

   

  Some others:

   

  Q( X(100) ) = REFLECTION OF X(3025) IN THE LINE X(1)X(2742)
  = (-a+b+c)*((b-c)*(2*a^2-(b+c)* a+(b-c)^2)*(a^2-2*(b+c)*a+b^2+ c^2))^2: : (barys)

  = on the incircle and on line {2348, 3021}

  = reflection of X(1358) in the line {X(1), X(142)}

  = reflection of X(3025) in the line {X(1), X(2742)}

  = reflection of X(3323) in the line {X(1), X(5519)}

  = [ 0.2629284986760934, 1.7965852893737350, 2.2755230514135970 ]

   

  
  Q( X(190) ) = REFLECTION OF X(3025) IN THE LINE X(1)X(2743)
  = (-a+b+c)*(3*a-b-c)^2*(b^2-4*b* c+c^2+(b+c)*a)^2*(b-c)^2 : : (barys)

  = on the incircle and on these lines: {11, 14112}, {3880, 6018}

  = reflection of X(11) in the line {X(1), X(13625)}

  = reflection of X(1357) in the line {X(1), X(474)}

  = reflection of X(3025) in the line {X(1), X(2743)}

  = reflection of X(14027) in the line {X(1), X(5516)}

  = [ 0.9397016102013350, 2.6563045336140130, 1.3679759846971390 ]

   

  César Lozada