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Concurrent circles and concurrent hyperbolas

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  • pneagoe
    Dear Hyacinthos, Let`s consider the triangle ABC and points Y, Z on the sidelines CA, AB. Let P be the point so that the angles
    Message 1 of 1 , Jan 24, 2007
      Dear Hyacinthos,

      Let`s consider the triangle ABC and points Y, Z on the sidelines
      CA, AB. Let P be the point so that the angles <AYP and <AZP are
      congruent.
      1.If the angles <AYP and <AZP have the same orientation then the
      geometric locus of P is the circumcircle Ca of the triangle AYZ
      2.If the angles <AYP and <AZP have different orientation then the
      geometric locus of P is a hyperbola Ha going through the points
      A, Y, Z.

      Now let's consider the triangle ABC, points X, Y, Z on the
      sidelines BC, CA, AB.
      1.If we have angles with the same orientation let Ca, Cb, Cc the
      circumcircles of the triangles AYZ, BZX, CXY. The circles Ca,
      Cb, Cc are concurrent in P. If P is on the circumcircle of the
      triangle ABC then X, Y, Z are collinear points.
      2.If we have angles with different orientation let Ha(going through
      A, Y, Z), Hb(going through B, Z, X), Hc(going through C, X, Y) be
      the hyperbolas. The hyperbolas Ha, Hb and Hc are concurrent in
      point P if and only if XYZ is the pedal triangle of the point P.
      If P is on the circumcircle of the triangle ABC then X, Y, Z are
      collinear points on the Simson's line of P (S(P)).

      Best regards,
      Petrisor Neagoe
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