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15580Re: A very simple geometry question..

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  • vprasad_nalluri
    Oct 1, 2007
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      --- In Hyacinthos@yahoogroups.com, "malisundaresan"
      <malisundaresan@...> wrote:
      > Prove that the line joining the -circumcenter of a triangle to a
      > vertex- is perpendicular to the line -joining the foot of
      altitudes of
      > adjoining sides.-
      > Some sort of a HINT please....
      Let E, F be the feet of the altitudes through
      B, C upon AC, AB respectively,
      Let X be the midpoint of AH (H orthocenter)
      X is the center of the circle through A, E, H, F.
      XE = XF
      The nine-point circle (N) passes through E and F
      NE = NF (each = R/2)
      Follows NX is the perpendicular bisector of EF.
      In the triangle AOH :
      X, N being the midpoints of AH, OH we have
      NX is parallel to OA.
      Together with the fact that NX is perpendicular to EF,
      it now follows that OA is perpendicular to EF

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