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equilateral triangle ABC

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  • orl_ml
    Given an equilateral triangle ABC and a point M in the plane (ABC). Let A , B , C be respectively the symmetric through M of A, B, C. a) Prove that there
    Message 1 of 2 , Mar 12, 2004
      Given an equilateral triangle ABC and a point M in the plane (ABC).
      Let A', B', C' be respectively the symmetric through M of A, B, C.

      a) Prove that there exists a unique point P equidistant from A and
      B', from Band C' and from C and A'.

      b) Let D be the midpoint of the side AB. When M varies (M does not
      coincide with D), prove that the circumcircle of triagle MNP (N is
      the intersection of the line DM and AP) pass through a fixed point.
    • ben_goss_ro
      ... For (a): We can see that triangle (B C A ) is obtained from (ABC) by a rotation of -60 degrees followed by a translation, and because if we compose a
      Message 2 of 2 , Apr 2, 2004
        --- In Hyacinthos@yahoogroups.com, "orl_ml" <orlando.doehring@h...>
        wrote:
        > Given an equilateral triangle ABC and a point M in the plane (ABC).
        > Let A', B', C' be respectively the symmetric through M of A, B, C.
        >
        > a) Prove that there exists a unique point P equidistant from A and
        > B', from Band C' and from C and A'.
        >
        > b) Let D be the midpoint of the side AB. When M varies (M does not
        > coincide with D), prove that the circumcircle of triagle MNP (N is
        > the intersection of the line DM and AP) pass through a fixed point.

        For (a): We can see that triangle (B'C'A') is obtained from (ABC) by
        a rotation of -60 degrees followed by a translation, and because if
        we compose a translation and a rotation we get a rotation, it means
        that triangle (B'C'A') is obtained from (ABC) by a rotation of -60
        deg, and the center of that rotation is the point we're looking for.
        Moreover, PAB', PBC" and PCA' are equilateral.

        For (b): Are you saying that M moves on a line? I drew some dynamic
        cketches and found that if M moves on a line l then that point moves
        on a line l' perpendicular to l. If M moves on a circle, then the
        circumcircle of MNP also moves on a circle.
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