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A conic centered at Euler line

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  • Francisco Javier
    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
    Message 1 of 4 , Oct 26, 2012
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      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
      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 2 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 3 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 4 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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