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[EMHL] Re: Line construction problem

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  • rafinad2003
    ... (P,s). ... Dear Antreas, As circle M* rolls around N, the base of the symmetry line moves along a pedal circle. When symmetry line passes through point P
    Message 1 of 6 , Jan 23, 2004
      --- In Hyacinthos@yahoogroups.com, "Antreas P. Hatzipolakis"
      <xpolakis@o...> wrote:
      > [APH]:
      > >> >Let P be a fixed point, and (M,R), (N,r) two given circles.
      > >> >
      > >> >Draw a line l through P such that the reflection (M*,R) of
      > >> >the circle (M,R) in the line l be tangent to the circle (N,r).
      > >>
      > >>
      > >> Now, instead of the zero circle (P,0) we are given the circle
      (P,s).
      > >>
      > >> Draw a line tangent to (P,s) such that....
      >
      > [Rafi]:
      >
      > > Find the homothety centers O' and O" of circle P with each of
      > > the two pedal circles corresponding to internal and external
      > > tangency of M* and N.
      > >
      > > Construct circles C1 and C2 on diameters MO' and NO".
      > >
      > > The intersections of the C1 with two pedal circles and
      > > the intersections of C2 with two pedal circles are the
      > > candidates to form together with points O' and O"
      > > the resulting symmetry axis tangent to circle P.
      > >
      > >
      > > Friendly,
      > >
      >
      > Dear Rafi.
      >
      > Probably only Darij could decipher the above !
      >
      > The general problem can be solved by the intersection of
      > a circle and a cubic.
      >
      > Antreas
      > --

      Dear Antreas,

      As circle M* rolls around N, the base of the symmetry line moves
      along a pedal circle. When symmetry line passes through point P
      the right angle made by the symmetry line and the center line MM*
      is inscribed in a circle that has segment MP (NP) as diameter.
      The cross-points of these circles give us the second point to
      construct line in question. There are two pedal circles -
      for internal and external tangency of M* and N.


      When P is a circle, it is reduced to a point at it's
      homothety centers with 2 pedal circles.

      Friendly,
      Rafi.
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