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945Where is the incenter?

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  • John Conway
    Jun 1, 2000
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      I was trying to prove for myself the assertion that the locus
      of the incenter over all triangles with given Euler segment is
      the interior of the orthocentroidal circle, or "arena". However,
      the algebra turns out to be a bit too tough even for me - I can
      prove that the incenter is always inside this circle, but not that
      all points of it are reached.

      So I hope that one of the hyacinths (Paul perhaps?) can help by
      finding the expression for the squared distance from the incenter
      to the midpoint of GH as a function of a,b,c.

      I'm also interested in the locus of the excenters. I thought
      at first that they would all lie outside the arena, but seem to
      have disproved this. There are two problems:

      1) what's the set of points where any excenter can be?

      2) supposing a > b > c, what are the individual loci of
      the a-, b-, and c- excenters?

      I suspect that the answer to 1) has a fairly simple shape,
      while those that arise in 2) might be quite complicated.

      Regards to all, John Conway
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