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Re: [EMHL] First 2010 locus

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  • Antreas
    Dear Paul I think that a generalization can be: Let ABCD be a quadrilateral and P a point. Which is the locus of P such that the circumcenters of
    Message 1 of 3 , Jan 1, 2010
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      Dear Paul

      I think that a generalization can be:

      Let ABCD be a quadrilateral and P a point.
      Which is the locus of P such that the circumcenters of
      PAB,PBC,PCD,PDA are concyclic?

      By using triangle geometry:

      We take ABC as reference triangle. Let the h. coordinates of
      the fixed point D be (p:q:r), and of the variable point P be
      (x:y:z). Then we find the equation of the locus in terms of
      the fixed p,q,r (and fixed elements of ABC).

      Note that we can take both D,P as variable points. For example:
      Let ABC be a triangle and P, P* two isogonal conjugate points
      (we took D = P*) etc

      > [APH]: Let ABC be a triangle, P a point, and O,Oa,Ob,Oc
      > the circumcenters of
      > ABC,PBC,PCA,PAB, resp. Which is the locus of P such
      > that O,Oa,Ob,Oc are concyclic?
      >
      > [PY] This is the case if and only if P is the on the circumcircle
      > of ABC. The four circumcenters coincide.

      Happy New Year

      APH
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