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A SHORT TUTORIAL ON VARIATIONAL GEOMETRY


(This tutorial is based on pictures taken from the screen, so quite big ones. I recommend to maximize your HELP window).

The "geometry - tool" panel can be seen clicking here


In this tutorial we will learn to create the EULER's line of a triangle. For those of you not very used to geometry or with a "not-very-strong" memory, I will remember that this line is the one that goes trough the orthocenter, the circumcenter and the mediacenter (?) of a triangle. This line goes trough the three points for every triangle and the relations between the two segments created, maintain the value 1/3 to 2/3.

We can create this figure just drawing three points by hand, all the rest being done trough "variational geometry".

Lets see the MG's window after drawing the three points: click here

As they are seen on the screen, just selected, we can click on the first button of the geometry-panel. Here is what we obtain with just one click.

The next step is to draw the line from a corner to the middle point of the opposite side. For that, we select one side and the point of the corner, as can be seen in this pix.

Then we click on the seventh button of our tools panel and we will see how a new line appears in MG's screen.

We continue with the same process for the other two sides and we obtain finally this drawing.

Once this three lines drown, we can continue with the drawing of the three mediatrix. This can be done at once selecting the three sides of our triangle, as in this picture, and clicking on the fourth button of our tools-panel our screen will look like this.

Now, lets create the intersection points on which we are interested. For doing so, first we select a couple of corresponding lines, as is shown here, and clicking on the eighth button our point is created. If selected, it can be clearly seen.

It must be remembered that, althought the lines are limited in length, their properties exist outside the "physical" segment. If the lines were shown in total length, crossing the screen, it would be extremely difficult to draw anything more that very simple things.

We have forgotten to create the intersection point of the first three lines. We do it just now, in the same way as it has been done with the mediatrix.

Althought I think that MG is a friendly user's program, I must accept that at this state there is a little bit of a mess whith our lines and points. The moment has come to use some colours to view our image more "crisply". This is done as usual and nothing special must be mentioned. Perhaps, it would be good to remember that if we want to give the same colour to several objects, the best way is to select the objects, group them, choose the colour and, finally, ungroup them. In this way it is needed only once to put the colour, instead of once per object. We can also choose the thickness of the majority of lines in a similar way.

We need to create a third internal point: the orthocenter. The preparation for the drawing of the first line, can be seen in this picture, and after clicking the fifth button, our perpendicular will appear.

Doing the same with the other two remaing lines, creating their point of intersection and joining the internal, extreme, intersection points, our screen will look like this one. Euler's line is there in dark red.

Well, now it's time to look at our drawing in its "real" aspect. So we can open the Ghostview screen and save our drawing. This is what appears in front of us. O.K. Yes, I agree... nice but not so impresive. Do not forget the fact that Hobby's MetaPost excell in many fields, and, specially, in quality. It is a picture of a vector-drawing what we have in front. Just increasing it 16X, we realize that the quality of our drawing is outstanding. If we increse it by 64X, we must accept that it is really impresive...

We have said, at the begining, that one of the points of Euler's line is the circumcenter. Let's take advantage of this fact and draw the circle trough the three coins. For doing that we must select the crossing point of the mediatrix-lines and one of the corner points. Then, just clicking on button tenth we will see this figure. Just to see the perfect match of our circle with the corners, I have increased the corresponding picture on ghostview by 64X,

Finally, it is important to understand that in this drawing we have created a few (quite a lot) of "internal-relations" that will be maintaned "forever and ever", until you close this MG's session. The origin of everything were three simple points place on the screen. Now, if we move anyone of those points, "everything will change... just to remain the same" (Lampedusa). In the next picture we see our drawing with one of those point selected. Then, I have maintained my finger for a short period over the "right-arrow.key" of my keyboard to obtain this new drawing. Finally, I have selected the corner at left and changed its position. I obtain another "version" of my drawing. And if there were some doubts about the extreme precission capability of this new feature, here is a final picture of this last drawing increased by 64X.


Notes. Sometimes, it can be tricky (not to say, clearly difficult) to select a line or a point in the middle of a big number of lines or other objects. When this is really difficult, there is no other "reasonable" way of doing that using the "double-arrow" button. It is important to know that the same effect obtained by clicking this button, is obtained by pressing the "q" key on the keyboard, and a lot quicker. I recommend it. Also, it is convenient to know that the last object "illuminated" is the easiest to select, but also the one-before-the-last is quite simple. And so on.

A few words must be said about the creation of the spiral-circle in those drawings. This circle is created selecting a couple of points. The circle will be drawn using one point as the center and another as the reference for the radius. So it is important to know wich one will be the center. The point that will be the center is the last one in the list that uses internally Java to control the objects. I understand that this is not a lot of information, but if you remember that selecting any object, grouping it (yes, alone) and ungrouping it, it is placed in this last position, the solution for our problem is simply. If not totally sure of having created and/or replaced the point to be the center as the last point, it is simply to select it, group and immediatly ungroup it, and the create the circle. Everything will be fine.

Finally, I will recommend to click in any part on the screen of MG after moving some object and before launching Ghostview and saving the drawing. Java is a wonder but as so many human wonders, it is "lazy"....

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