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Re: [EMHL] Miquel Puzzle [SPOILER]

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  • Antreas P. Hatzipolakis
    ... The solution is based in two lemmata (well-known theorems): Lemma #1 : The orthocenters H(123), H(234), H(341), H(412) of the four triangles (123), (234),
    Message 1 of 1 , Jun 1, 2002
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      On 1-06-02, Antreas P. Hatzipolakis <xpolakis@...> wrote:

      >Let 1,2,3,4 be four lines forming the
      >complete quadrilateral (1234).
      >
      >The Miquel point of (1234) is the common point
      >of the circles (123), (234), (341), (412)
      >[and also the circle (O(123)O(234)O(341)O(412))
      >ie the circle passing through the four O's of the
      >triangles (123), (234), (341), (412)]
      >
      >Now, define the Miquel point of (1234)
      >by using only LINES; not circles.

      The solution is based in two lemmata (well-known theorems):

      Lemma #1 :
      The orthocenters H(123), H(234), H(341), H(412)
      of the four triangles (123), (234), (341), (412)
      are collinear (Steiner Line of (1234)).

      Lemma #2:
      Let P be a point. The reflections of the line
      HP in the sidelines of triangle ABC concur in
      a point on the circumcircle of ABC.

      ==>

      The Miquel Point of the complete quadrilateral
      (1234) is the point of concurrence of the reflections
      of the Steiner line of (1234) in its sidelines 1,2,3,4.

      Antreas
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