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14220Special Sondat

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  • Francois Rideau
    Oct 1, 2006
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      Dear friends
      I have noticed that the Nikos configuration in message 6466 was simply the
      figure of the special Sondat theorem that is to say 2 orthologic triangles
      ABC and A'B'C' with the same orthology center O.
      Then :
      1°ABC and A'B'C' are perspective wrt some perspector S
      2°Line SO is orthogonal to the perspectrix.
      Nikos only prove the 1st point and I have said I don't like his proof very
      much.
      In the next message n°6467, Darij noticed that ABC and A'B'C' are conjugate
      wrt some circle of center O and that implies the 1st point (and also the 2st
      as we shall see).
      But Darij regretted this proof was not valid if the circle is not real.
      I don't agree with him for polarity is in fact defined wrt a quadratic form
      before beeing defined wrt the conic possibly vanishing the form.
      So Darij proof is valid in all cases.
      Now if we come back to the general situation of 2 triangles ABC and A'B'C'
      conjugate wrt some conic {Gamma}, then it is well known that ABC and A'B'C'
      are perspective with perspector S and perspectrix L. Moreover L is the polar
      line of S wrt conic {Gamma}.
      In the special Sondat configuration, as {Gamma} is a circle of center O and
      L is the polar line of S wrt circle {Gamma}, then line SO is orthogonal to L
      and we are done with the 2st point.
      So I think Darij proof is The Synthetic Solution of the special Sondat
      Theorem.
      Friendly
      François


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