## Re: Sherman's fourth side of a triangle

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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
Message 1 of 2 , Jul 19, 2012
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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