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17327Re: NPC and Kiepert hyperbola

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  • chris.vantienhoven
    Mar 1 10:28 AM
      Dear Francisco.

      I understand what you are saying.
      It is an interesting question.
      Everything is allright for me. Except for your statement "The line
      PQR here is the parallel to DP', LP", etc. through P." That is just
      the point you started with. Under what conditions is there a line PQR?
      For a better understanding let us name:
      • Line L = the line through Q and R
      • Pi = Infinity Point, being the IntersectionPoint of parallel
      lines DP', LP'', etc.
      We started with the question under what conditions P,Q,R are
      collinear?
      You noticed that when P is a point on the Kiepert Hyperbola that in
      that case P,Q,R are collinear.
      This is easy to be confirmed with drawings.
      Now you state that when P is on NPC, then Q and R coincide with
      Infintypoint Pi.
      Then you state (in other words): so P.Q.R = P.Pi.Pi = P.Pi, ergo they
      are collinear.
      Is that right?
      However the question is not to draw the line P.Pi. The original
      question is to draw a line through Pi.Pi and then look if this line
      coincides with P.
      When you draw a line through Pi.Pi you actually draw a line through
      Pi (evident) but you don't know if it comes from the right or the
      left side of the plane.
      It can be any line parallel to lines DP', LP'', etc. There is an
      endless amount of these lines possible.
      As soon as we found that very special line that is inherent to the
      problem then we have to review if P is on this line.
      Of course this should be confirmed with drawings (as in real life we
      should try to confirm all things that are said to be true).
      When I make drawings in Cabri and take a variable point P and move it
      to any point on NPC I see what happens to QR in these drawings:
      1. It is moving to a parallel line to DP', LP'', etc.
      2. It is going through the point T on the BrocardAxis that I
      described in earlier mail.
      3. It is NOT going through P !
      So this is a confirmation of the contrary of your statement. For me
      it is clear that P,Q,R are not collinear when P is on NPC, except for
      the situation when it is an intersectionpoint of NPC with the Kiepert
      Hyperbola.
      Best regards,
      Chris van Tienhoven

      p.s. There is a comparable situation when you take 2 points Q and R
      on a circle and any point P outside the circle and you want to know
      when P,Q,R are collinear. As soon as Q and R coincide you notice that
      in this case (inherent to the problem) QR is the tangent to the
      circle and P is not on the tangent.
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