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Re: [EMHL] Re: Locus (eguilateral triangles)

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  • Nikolaos Dergiades
    Dear Antreas, Angel, Francisco I think that If L1,L2,L3 are the peprendicular bisectors of the sides of ABC (BC, CA, AB) and the lines 1, 2, 3 pass through G
    Message 1 of 7 , Jan 24, 2013
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      Dear Antreas, Angel, Francisco

      I think that

      If L1,L2,L3 are the peprendicular
      bisectors of the sides of ABC (BC, CA, AB)
      and the lines 1, 2, 3 pass through G
      then
      La is the reflection of L1 in 1
      Lb is the reflection of L2 in 2
      Lc is the reflection of L3 in 3.
      The lines are concurrent at Q and the
      locus of Q is the circle with center G
      that passes through the first Fermat point
      and not the circumcenter of ABC.

      Is this true or false?

      Best regards
      Nikos Dergiades



      > Dear Francisco Javier,
      >
      > Indeed, through the circumcenter and therefore by their
      > antipodal, X(381). These points correspond to the lines
      > 1  parallel to the Euler line and its perpendicular
      > direction, respectively.
      >
      > Best regards,
      >
      > Angel M.
      >
      >
      > --- In Hyacinthos@yahoogroups.com,
      > "Francisco Javier"  wrote:
      > >
      > > Dear Angel,
      > >
      > > this circle goes through the circumcenter!
      > >
      > > Best regards,
      > >
      > > Francisco Javier.
      > >
      > > --- In Hyacinthos@yahoogroups.com,
      > "Angel"  wrote:
      > > >
      > > > Dear Antreas
      > > >
      > > > For every line 1, the lines La, Lb and Lc are
      > concurrent. The intersection point is on the circle (with
      > center the centroid) of the barycentric equation:
      > > >
      > > > CiclicSum [ a^2y*z + x(x+y+z)
      > (a^4(b^2+c^2)-a^2(2b^4+b^2c^2+2c^4)+(b^2-c^2)^2(b^2+c^2))/
      > (3(-a+b+c)(a+b-c)(a-b+c)(a+b+c)) ]=0.
      > > >
      > > >
      > > > Angel M.
      > > >
      > > > --- In Hyacinthos@yahoogroups.com,
      > "Antreas"  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
      > > > >
      > > >
      > >
      >
      >
      >
      >
      > ------------------------------------
      >
      > Yahoo! Groups Links
      >
      >
      >     Hyacinthos-fullfeatured@yahoogroups.com
      >
      >
    • Antreas Hatzipolakis
      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
      Message 2 of 7 , Jan 24, 2013
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        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]
      • 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 3 of 7 , Jan 24, 2013
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          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]
          >
          >
          >
          > ------------------------------------
          >
          > Yahoo! Groups Links
          >
          >
          >     Hyacinthos-fullfeatured@yahoogroups.com
          >
          >
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