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Re: [EMHL] Re: Rhombi problem

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  • xpolakis@otenet.gr
    Dear Floor and Paul ... This circle is the Conway Circle of the medial triangle. See the thread: http://forum.swarthmore.edu/epigone/geom.puzzles/wilyangdimp/
    Message 1 of 1 , Apr 30, 2001
      Dear Floor and Paul

      [FvL]:
      >>Let ABC the a triangle. Draw externally on each side a semicircle. If
      >>we take two sides of the medial triangle, and extend these so that
      >>they each meet two of the semicircles we get four points on these
      >>semicircles. There is a rhombus tangent to the three semicircles in
      >>these four points. Find the center of the rhombus.

      [PY]:
      >This is the Spieker center of triangle ABC, which is the incenter of
      >the medial triangle. In fact, if we consider the six points formed by
      >extending all three sides of the medial triangles, the 6 points lie
      >on a circle. This circle has the Spieker point as center, radius
      >(1/2)sqrt(r^2+s^2), and is orthogonal to each of the three excircles
      >of triangle ABC.


      This circle is the Conway Circle of the medial triangle.

      See the thread:

      http://forum.swarthmore.edu/epigone/geom.puzzles/wilyangdimp/

      I append below JHC's first posting in the thread.

      Antreas

      __________________________________________________________________


      Subject: a nice little theorem
      From: John Conway <conway@...>
      Date: Mon, 16 Mar 1998 15:06:05 -0500 (EST)


      Here's a little theorem I found about a week ago - I don't
      know if it's new:

      Produce the edges of a triangle ABC to distances

      a beyond A, b beyond B, c beyond C

      where a,b,c are the edgelenths of the triangle. Then the 6 points
      so constructed lie on a circle:
      |
      | /
      | /
      | /
      |/
      B
      /|
      / |
      / |
      / |
      / |
      / |
      / |
      ------------C-------A----------------
      / |
      / |
      / |
      / |
      / |
      / |
      |
      | John Conway
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