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Re: [EMHL] Locus (eguilateral triangles)

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  • Nikolaos Dergiades
    Dear Antreas, Yes the lines are concurrent. If in barycentrics the point at infinity of line 1 is the point (1 : x :-1-x) of line 2 is the point (1 : y :-1-y)
    Message 1 of 7 , Jan 24 3:46 PM
      Dear Antreas,
      Yes the lines are concurrent.
      If in barycentrics the point at infinity
      of line 1 is the point (1 : x :-1-x)
      of line 2 is the point (1 : y :-1-y)
      of line 3 is the point (1 : z :-1-z)
      then the lines La, Lb, Lc are concurrent at the point

      Q = {-x^4 y + x^3 y^2 - x^3 y^3 + x y^4 + 2 x^2 y^4 - x^4 z + 6 x^3 y z -
      2 x^4 y z - 6 x^2 y^2 z + 5 x^3 y^2 z - 3 x y^3 z - 5 x^2 y^3 z -
      2 x y^4 z + x^3 z^2 - 6 x^2 y z^2 + 5 x^3 y z^2 + 12 x y^2 z^2 -
      6 x^2 y^2 z^2 - y^3 z^2 + 6 x y^3 z^2 + y^4 z^2 - x^3 z^3 -
      3 x y z^3 - 5 x^2 y z^3 - y^2 z^3 + 6 x y^2 z^3 - 4 y^3 z^3 +
      x z^4 + 2 x^2 z^4 - 2 x y z^4 + y^2 z^4,
      x^4 y^2 - 2 x^3 y^3 + 2 x^2 y^4 + x^3 y^4 - 4 x^4 y z + 6 x^3 y^2 z -
      x^4 y^2 z - 5 x^2 y^3 z - 3 x^3 y^3 z - x y^4 z + x^4 z^2 +
      6 x^3 y z^2 - x^4 y z^2 - 6 x^2 y^2 z^2 + 12 x^3 y^2 z^2 +
      5 x y^3 z^2 - 6 x^2 y^3 z^2 + x y^4 z^2 - 2 x^3 z^3 - 5 x^2 y z^3 -
      3 x^3 y z^3 + 5 x y^2 z^3 - 6 x^2 y^2 z^3 - 2 y^3 z^3 +
      6 x y^3 z^3 - y^4 z^3 + 2 x^2 z^4 + x^3 z^4 - x y z^4 + x y^2 z^4 -
      y^3 z^4, x^4 y - x^3 y^2 + 2 x^4 y^2 - 3 x^3 y^3 - x y^4 - x^2 y^4 -
      x^3 y^4 + x^4 z - 6 x^3 y z + 6 x^2 y^2 z - 5 x^3 y^2 z +
      x^4 y^2 z + 3 x y^3 z + 16 x^2 y^3 z + 3 x^3 y^3 z - 3 x y^4 z -
      x^3 z^2 + 2 x^4 z^2 + 6 x^2 y z^2 - 5 x^3 y z^2 + x^4 y z^2 -
      12 x y^2 z^2 - 6 x^2 y^2 z^2 - 12 x^3 y^2 z^2 + y^3 z^2 -
      5 x y^3 z^2 + 6 x^2 y^3 z^2 + 2 y^4 z^2 - x y^4 z^2 - 3 x^3 z^3 +
      3 x y z^3 + 16 x^2 y z^3 + 3 x^3 y z^3 + y^2 z^3 - 5 x y^2 z^3 +
      6 x^2 y^2 z^3 - 6 x y^3 z^3 + y^4 z^3 - x z^4 - x^2 z^4 - x^3 z^4 -
      3 x y z^4 + 2 y^2 z^4 - x y^2 z^4 + y^3 z^4}

      Nikos

      > I am wondering about a
      > generalization. One possibly is this:
      >
      > Let 1,2,3 be three lines (concurrent with given angles or
      > bounding a
      > triangle).
      >
      > Let La, Lb, Lc be the lines joining the CENTROIDS of the
      > respective
      > triangles.
      >
      > Are always these lines concurrent??
      >
      > Let's consider a special triad of such lines, namely:
      >
      > 1,2,3 = the reflections of the Euler line in the sidelines
      > BC,CA,AB, resp.
      > (concurrent on the circumcircle).
      >
      > Denote:
      >
      > a1, a2, a3 = the lines through A parallel to 1,2,3,resp.
      > b1, b2, b3 = the lines through B parallel to 1,2,3,resp.
      > c1, c2, c3 = the lines through C parallel to 1,2,3,resp.
      >
      > La = the line joining the centroids of the triangles bounded
      > by the lines
      > (a1,b2,c3) and (a1,b3,c2)
      > [the triangle (a1,b2,c3) is a point, which we take as
      > centroid]
      >
      > Similarly Lb, Lc.
      >
      > Are tle lines La,Lb,Lc concurrent?
      >
      > APH
      >
      >
      >
      > On Thu, Jan 24, 2013 at 1:18 PM, Antreas <anopolis72@...>
      > wrote:
      >
      > > **
      > >
      > >
      > > Let ABC be a triangle, 1 a line and 2,3 two lines such
      > that
      > > the triangle bounded by the lines (1,2,3) is
      > equilateral.
      > >
      > > Denote:
      > >
      > > a1, a2, a3 = the lines through A parallel to
      > 1,2,3,resp.
      > > b1, b2, b3 = the lines through B parallel to
      > 1,2,3,resp.
      > > c1, c2, c3 = the lines through C parallel to
      > 1,2,3,resp.
      > >
      > > La = the line joining the centers of the equilateral
      > > triangles bounded by the lines (a1,b2,c3) and
      > (a1,b3,c2)
      > >
      > > Lb = the line joining the centers of the equilateral
      > > triangles bounded by the lines (b1,c2,a3) and
      > (b1,c3,a2)
      > >
      > > Lc = the line joining the centers of the equilateral
      > > triangles bounded by the lines (c1,a2,b3) and
      > (c1,a3,b2)
      > >
      > > If 1 is the trilinear polar of point P, which is the
      > > locus of P such tah La,Lb,Lc are concurrent?
      > >
      > > APH
      > >
      >
      >
      > [Non-text portions of this message have been removed]
      >
      >
      >
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