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Re: A conic centered at Euler line

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  • Francisco Javier
    Sorry for the character codes, See http://garciacapitan.blogspot.com.es/2012/10/a-conic-centered-at-euler-line.html for a more readable version
    Message 1 of 4 , Oct 26, 2012
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      Sorry for the character codes,

      See

      http://garciacapitan.blogspot.com.es/2012/10/a-conic-centered-at-euler-line.html

      for a more readable version

      --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:
      >
      > I found the following:
      >
      > Theorem. Given a triangle ABC, call Γ the locus of points P such that polar of P with respect to the circumcircle is tangent to the nine point circle. Then we have:
      > 1) Γ is a conic whose center is X26, the circumcenter of the tangential triangle.
      > 2) Γ is an ellipse, parabola o hyperbola if and only if the triangle is acute, triangle or obtuse.
      > 3) The diameter on the transverse axis of the conic is also a diameter of the circumcircle of tangential triangle. Therefore, the transverse axis of the conic is the Euler line of the triangle.
      > 4) If α, β are the lengths of the transverse and conjugate axes of the conic respectively, we have the relation
      >
      > (αβ)^2=1−OH^2/R^2
      >
      > 5) The foci of the conic are O and O′, where O′ is the reflection of O on the center X26.
      >
    • Francisco Javier
      Dear friends: I now see that this is a particular case of some projective theorem: the locus of poles with respect a conic of the tangents to another conic is
      Message 2 of 4 , Oct 26, 2012
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        Dear friends:

        I now see that this is a particular case of some projective theorem: "the locus of poles with respect a conic of the tangents to another conic is also a conic".

        Here is the version for two circles:

        (A) and (B) are circles
        The line AB intersect (B) at M and N
        M' and N' are the inverses of M and N with respect to (A)
        J is the inverse of A with respect to (B)
        O is the inverse of J with respect to (A)
        A' is the reflection of A on O
        The locus points P such that the polar of P with respect to (A) is tangent to (B) is a conic with foci A and A' and diameter M'N'.

        what is the description of the locus in the general case in terms of the two given conics?

        Thank you.
      • luis240985
        ... Dear Francisco, it is the polar conic of one conic with respect to another, in other words, the dual of a conic is a conic. For further properties you can
        Message 3 of 4 , Nov 4, 2012
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          --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:
          >
          > Dear friends:
          >
          > I now see that this is a particular case of some projective theorem: "the locus of poles with respect a conic of the tangents to another conic is also a conic".
          >
          > Here is the version for two circles:
          >
          > (A) and (B) are circles
          > The line AB intersect (B) at M and N
          > M' and N' are the inverses of M and N with respect to (A)
          > J is the inverse of A with respect to (B)
          > O is the inverse of J with respect to (A)
          > A' is the reflection of A on O
          > The locus points P such that the polar of P with respect to (A) is tangent to (B) is a conic with foci A and A' and diameter M'N'.
          >
          > what is the description of the locus in the general case in terms of the two given conics?
          >
          > Thank you.
          >

          Dear Francisco, it is the polar conic of one conic with respect to another, in other words, the dual of a conic is a conic. For further properties you can see Geometry of conics by A.V. Akopyan and A.A. Zaslavsky, pages 70-72.
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