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Sherman's fourth side of a triangle

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  • Chris Van Tienhoven
    Dear friends, I made some observations regarding the Sherman s fourth side of a triangle. 1. These 2 ETC-points lie on Sherman s fourth side of a triangle:
    Message 1 of 2 , Jul 19, 2012
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      Dear friends,

      I made some observations regarding the Sherman's fourth side of a triangle.

      1. These 2 ETC-points lie on Sherman's fourth side of a triangle:
      X(3259) = Crosssum of X(100) and X(104)
      X(3326) = Incircle Transform T(X9909))

      2. There is a 3rd non-ETC-point P3 that lies on Sherman's fourth side of a triangle, which is the intersection point X(1).X(3) ^ X(11).X(123).
      First Baricentric Coordinate: a(a-b-c)(b-c)^2(a^2-b^2-c^2)(b^3+2abc+c^3-a^2(b+c)-bc(b+c))
      Search: 3.2704336425
      It also lies on lines X(1320).X(1809), X(1364).X(3270), X(3259),X(3326).

      3. In the most recent Forum Geometricorum paper of Paul Yiu about this subject there is a point P mentioned in the 2nd construction at page 224. This point P is X(953).

      4. The Midpoint (Q in Paul Yiu's figure 4)of P=X(953) and X(4) is X(3259) and lies as mentioned above on Sherman's fourth side of a triangle.
      X(3259) is also the Center of the rectangular Hyperbola through A,B,C,X(4),P=X(53).

      5. The point of tangency (T in Paul Yiu's figure 4) of the Incircle and Sherman's fourth side is X(3326).

      Best regards,

      Chris van Tienhoven


      >Sherman's paper is available in pdf format here

      >http://poncelet.math.nthu.edu.tw/disk5/js/cardioid/9.pdf

      >Note that "THE FOURTH SIDE OF THE TRIANGLE" was the title of a novel
      written by
      >Ellery Quenn. See a short note about here

      >http://www.shvoong.com/books/mystery-and-thriller/1683156-fourth-triangle/

      >APH

      On Wed, Jul 18, 2012 at 4:46 PM, forumgeom forumgeom <ForumGeom@...>wrote:

      > **
      >
      >
      > The following paper has been published in Forum Geometricorum. It can be
      > viewed at
      >
      > http://forumgeom.fau.edu/FG2012volume12/FG201220index.html
      >
      > The editors
      > Forum Geometricorum
      >
      > Paul Yiu, Sherman's fourth side of a triangle,
      > Forum Geometricorum, 12 (2012) 219--225.
      >
      > Abstract. We give two simple ruler-and-compass constructions of the line
      > which, like the sidelines of the triangle, is tangent to the incircle and
      > cuts the circumcircle in a chord with midpoint on the nine-point circle.
      >
      >

      [Non-text portions of this message have been removed]
    • Chris Van Tienhoven
      Dear Friends, I was a bit quick with my former message. I noticed that X(3259,X(33260, X(953) already are mentioned in the paper of Paul Yiu. I am sorry for
      Message 2 of 2 , Jul 19, 2012
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        Dear Friends,

        I was a bit quick with my former message.
        I noticed that X(3259,X(33260, X(953) already are mentioned in the paper of Paul Yiu. I am sorry for bothering you with this.
        I hope the rest of my notes (2.) are interesting enough.
        Best regards,

        Chris van Tienhoven

        --- In Hyacinthos@yahoogroups.com, "Chris Van Tienhoven" <van10hoven@...> wrote:
        >
        > Dear friends,
        >
        > I made some observations regarding the Sherman's fourth side of a triangle.
        >
        > 1. These 2 ETC-points lie on Sherman's fourth side of a triangle:
        > X(3259) = Crosssum of X(100) and X(104)
        > X(3326) = Incircle Transform T(X9909))
        >
        > 2. There is a 3rd non-ETC-point P3 that lies on Sherman's fourth side of a triangle, which is the intersection point X(1).X(3) ^ X(11).X(123).
        > First Baricentric Coordinate: a(a-b-c)(b-c)^2(a^2-b^2-c^2)(b^3+2abc+c^3-a^2(b+c)-bc(b+c))
        > Search: 3.2704336425
        > It also lies on lines X(1320).X(1809), X(1364).X(3270), X(3259),X(3326).
        >
        > 3. In the most recent Forum Geometricorum paper of Paul Yiu about this subject there is a point P mentioned in the 2nd construction at page 224. This point P is X(953).
        >
        > 4. The Midpoint (Q in Paul Yiu's figure 4)of P=X(953) and X(4) is X(3259) and lies as mentioned above on Sherman's fourth side of a triangle.
        > X(3259) is also the Center of the rectangular Hyperbola through A,B,C,X(4),P=X(53).
        >
        > 5. The point of tangency (T in Paul Yiu's figure 4) of the Incircle and Sherman's fourth side is X(3326).
        >
        > Best regards,
        >
        > Chris van Tienhoven
        >
        >
        > >Sherman's paper is available in pdf format here
        >
        > >http://poncelet.math.nthu.edu.tw/disk5/js/cardioid/9.pdf
        >
        > >Note that "THE FOURTH SIDE OF THE TRIANGLE" was the title of a novel
        > written by
        > >Ellery Quenn. See a short note about here
        >
        > >http://www.shvoong.com/books/mystery-and-thriller/1683156-fourth-triangle/
        >
        > >APH
        >
        > On Wed, Jul 18, 2012 at 4:46 PM, forumgeom forumgeom <ForumGeom@...>wrote:
        >
        > > **
        > >
        > >
        > > The following paper has been published in Forum Geometricorum. It can be
        > > viewed at
        > >
        > > http://forumgeom.fau.edu/FG2012volume12/FG201220index.html
        > >
        > > The editors
        > > Forum Geometricorum
        > >
        > > Paul Yiu, Sherman's fourth side of a triangle,
        > > Forum Geometricorum, 12 (2012) 219--225.
        > >
        > > Abstract. We give two simple ruler-and-compass constructions of the line
        > > which, like the sidelines of the triangle, is tangent to the incircle and
        > > cuts the circumcircle in a chord with midpoint on the nine-point circle.
        > >
        > >
        >
        > [Non-text portions of this message have been removed]
        >
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