Open source cross-platform software of geometry, analysis and simulation - Yves Biton
 
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High school level example 3 (Java version) : Barycentric transformation and use of constructions.

publication Sunday 16 January 2011.


I want to illustrate here a new theoreme I discovered and demonstrated a few years ago. If you are interested in the demonstration you can have a look at this address but it is written in french ...

Our goal is to illustrate a few properties of the transformation that, given three points A, B et C, associates to a point M the barycenter point of balanced points (A ; MA), (B; MB) et (C; MC).

We will create a new Construction that will create the point image of a point M through this transformation.

Create a new figure with MathGraph32Java (it must have a length unity).

Create now four free points (icon ) and name these points A, B, C and M (icon ).

Use tool to measure lengthes MA, MB and MC (measure tools are of one time use, so use key F9 to reactivate the last tool used).

Let’s now create the barycenter with tool . Click first on point A. A dialog box pops up to enter the coefficient of point A. Use button Values and choose MA for coefficient then validate by OK. In the same way, give MB and MC for coefficients to points B and C then right click to get the barycenter point created. Name this point M’.

We can now create our construction.

Use menu item Constructions - Sources objects choice - Graphical and click on A, B, C and M.

Now use menu item Constructions - Final elements choice - Graphical and click on the barycenter point M.

We must now save our construction on a disk file. Use menu item Construction - Save construction of figure to file (keeping the same name for the file and the construction is a good choice).

Now create two free points ((icon ) , a circle with the first point as center and the second point on the circle ((icon ) ) .

Create a point linked to this circle (icon ) and name this point N.

Use menu item Construction - Implement construction from file. A dialog box pops up. The only construction avilable is already selected. Validate with OK.

The indication line asks for a click on the first point. Click on A.
In the same way, click on B then on C. Click last on point N to get its image through the transformation. A new point appears.

In the color palette, activate the blue color.

We will now create the point locus of this last point generated through the positions of point N on the circle.

For this click on icon .

Click first on the last point created (image point of N) then on point N (don’t forget to read the instructions on the indication line).
A dialog box pops up. Ask for 100 points and select the Closed locus check box. Validate.

Use menu item Constructions - Sources elements choice - Graphical and click on A, B, C then on the circle.

Use menu item Constructions - Final elements choice - Graphical and click on the point locus you just created.

We must now finalize the construction with menu item Constructions - Finish current construction.

Fill in the dialog box as shown below :

Now save this construction in a file on your system through menu item
Construction - Save construction of figure to a file. Please remeber the directory used to save this mgc file.
We wiil now use this last construction in another MathGraph32 figure.

Create a new figure with a length unity which is the dafault choice).

Create two points and a circle with center the first point and goieng through the second one (icon ).

Name the center O.

Use icon to create thre points linked to this circle. Jojn these three points with segments ().

Use menu item Constructions - Incorporate construction in figure from file and open transformation named CircleImage from the file directory where you saved it.

With menu item Constructions - Implement construction from figurre, open construction CircleImage.

Click first on points A, B and C then on the circle.

The image point locus appears. This locus seems to be a triangle (and it is by my theoreme).

We will now illustrate the image of the inside of the triangle ABC through this transformation.

Create the segment [OA] (icon ) and a point linked to this segment we will call P.

Now create the circle with center O and through point P (icon ).

With menu item Constructions - Implement construction from file, open construction Circle image.

Click on points A, B and C then on the circle of center O and going through P. The image of the circle appears.

Let’s now create the object locus of this point locus generated through the positions of point P on segment [].

For this, use menu item Create - Object locus - Generated through linked point. Click first on the last point locus created the on point P.
In the dialog box popping up, ask for 25 objects. The object locus is created.

Please notice that you can change the color of the object locus but not the line style which is the line style of the locus point ti was generated with.

Here is below the figure animated by MathGraph32 JavaScript motor.

You can capture A, B, C and P.