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Re: [EMHL] Poncelet points on the Euler line

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  • Antreas
    Dear Randy Thanks!! I think it would be interesting to study this special case: Let p(pN) be the point where concur the NPCs of N1N2N3, pNN1N2, pNN2N3, pNN3N1
    Message 1 of 3 , Feb 11, 2013
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      Dear Randy

      Thanks!!

      I think it would be interesting to study this special case:

      Let p(pN) be the point where concur the NPCs of N1N2N3, pNN1N2,
      pNN2N3, pNN3N1 (ie we replace N with pN).

      Is it interesting ? ie ls it lying on some interesting curves, lines?

      In my figure it lies on the circle centered at N with radius NppN

      APH

      [Randy Hutson]
      > pP is the center of the rectangular circumhyperbola through P.
      > ppP does not, in general, lie on the Euler line.
      >
      > Some results:
      >
      > P=X(1), the NPCs are concurrent, with center = non-ETC 1.121590125545969 (on lines 1,5 3,962).
      > P=X(2), ppP=non-ETC 1.690358502447462
      > P=X(3), ppP=X(140)
      > P=X(4), ppP=undefined
      > P=X(5), ppP=X(3628)
      > P=X(6), ppP=non ETC 0.780037257060191
      > P=X(7), ppP=non ETC 0.750876768572663
      > P=X(8), ppP=non ETC 2.966801160450799
      > P=X(9), ppP=non-ETC 0.972023454564163
      > P=X(10), ppP=non-ETC 2.238481946743318
      > P=X(20), ppP=non-ETC 6.363850996796102
      > P=X(21), ppP=non-ETC -1.717011738240629
      > P=X(22), ppP=non-ETC -4.036288926987237
      >
      > Of these, only X(140) and X(3628) lie on the Euler line. 
      > The ppP for points P on the Euler line do not even lie on the
      > same conic. Locus?
      [APH]
      > >Let ABC be a triangle, (N),(N1),(N2),(N3) the NPCs of
      > >ABC,NBC,NCA,NAB, resp. [concurrent at pN]. The NPCs of the
      > >triangles N1N2N3, NN1N2, NN2N3, NN3N1 concur at point ppN
      > >on the Euler Line of ABC.
      > >
      > >Coordinates of ppN?
      > >
      > >Generalization:
      > >
      > >Let ABC be a triangle, P a point, (N),(N1),(N2),(N3) the NPCs
      > >of ABC, PBC, PCA, PAB resp. [concurrent at pP]. If P is on
      > >the Euler line of ABC, then the NPCs of N1N2N3, PN1N2, PN2N3,
      > >PN3N1 concur at point ppP on the Euler Line of ABC.
      > >
      > >True??
      > >
      > >APH
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