- 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

written by

>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

>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] - 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]

>