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Re: A locus problem

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  • wolkb@cc.umanitoba.ca
    ... ABCP. [snip] ... That s the locus Antreas asked for. The locus of the center of this reflection (as P varies on the circumcircle) is also interesting,
    Message 1 of 6 , Feb 28, 2001
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      jean-pierre.ehrmann wrote:
      > Dear Antreas, John and other Hyacinthists
      > [Antreas]
      > > > Let ABC be a triangle and P a point on its circumcircle.
      > > > It is well known
      > > > that the four orthocenters Hi, i = 1,2,3,4, of the triangles
      > > > ABP, PAC, CPB, CAB, resp.,form a quadrilateral congruent to
      ABCP.

      [snip]
      > If Ha is the orthocenter of PBC,... the segments AHa, BHb, CHc, PH
      > have the same midpoint.
      >
      > Hence the congruence is the reflection w.r.t. this point and the
      > locus of the the circumcenter of (Ha,Hb,Hc,H) is the circle with
      > center H, radius R.
      > Best regards. Jean-Pierre

      That's the locus Antreas asked for. The locus of the center of this
      reflection (as P varies on the circumcircle) is also interesting,
      because it is a trivial result that the midpoint of PH lies on the
      nine-point circle.

      Barry Wolk <wolkb AT ccu.umanitoba.ca>
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