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21269Re: A conic centered at Euler line

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  • luis240985
    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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