You can find all the files listed here in the standard Geometer release, but if you just loaded the executable or have an old release, the files are also available for download from this page.
The classifications of "elementary", "intermediate", and "advanced" below refer not to the difficulty of the mathematical concept illustrated, but rather with the familiarity you should have with the Geometer program to use the diagram effectively.
There are various things that can be done, depending on the diagram:
For Advanced Users Only: Advanced users of Geometer can edit the text version of the diagram (using the "Edit Geometry" pull-down menu command) and modify some of the numbers that control the relative sizes of the "wheels" and the relative number of "teeth" on the gears. It's easy to mess things up, but here's something that works:
Change the lines
v1 = .f.rpn(25.000000);
v2 = .f.rpn(5.000000);
v1 = .f.rpn(28.000000);
v2 = .f.rpn(7.000000);
Make the change and press "Run Script" again.
Notice that the wedges that made up the circle are simply rearranged to form something that looks a lot like a rectangle, and if the more edges there are, the more like a rectangle it would look. Also, remember that the circumference of the original circle is 2πr, so one side of the "rectangle" will be approximately πr and the other will be approximately r. Thus the area of the rectangle will be approximately πr².
Each of the trigonometric functions is illustrated by the length of a segment, and each of the six segments is a different color. As the angle moves around for 360 degrees, it's easy to see why functions like sine and cosine are restricted to be between -1 and 1, and why certain functions "go to infinity" for certain angles.
Of course, you can press the "Run Script" button as many times as you like.
Load the file Cvvvvv.T into Geometer, and note that the five points lie on an ellipse. Move the points to see how the associated conic section changes. Notice that certain configurations (three points in a line, for example) do not admit a conic section passing through them, or at best, admit only a degenerate conic section.
You can modify the figure by sliding the red and blue points on the line in the upper-left of the figure before pressing the "Run Script" button. They control the relative speeds of the red and blue points moving around the circle. Their position is rounded to an integer, so tiny movements may have no effect on the drawing. If they are set to the same position then the two points will move at the same speed, and you'll get the most boring lissijous figure; a circle.
The harmonic series is: 1, 1/2, 1/3, 1/4, 1/5, et cetera. If the length of the original rectangle in the figure is 1, successive points are placed at lengths of 1/2, 1/3, and so on from the left end of the rectangle.
Note that a description of each step appears at the bottom of the drawing, and note also that at any point, the relative sizes of the circles can be modified. What happens if the circles intersect? How about if one lies completely within the other?
Although it is not proved in this diagram, the center of the nine-point circle (see Nine-Point Circle below) also lies on the Euler line.
Poncelet's theorem states that if one circle is completely inside another, and if you start at a point on the outer circle, construct a tangent to the inner circle, and find where that line intersects the outer circle again, and then repeat the process over and over, if the line comes back to the original point, then it will come back no matter what starting point you choose.
This figure illustrates a pair of such circles where the line does happen to return after exactly four steps. Move the center of the inner circle and its diameter is automatically recalculated to the right size to make the line come back. Then move the point labeled "A" on the outer circle to see that no matter what the starting point, the series of lines closes.
Load the Geometer diagram Peaucellier.T and move the point labeled A around on the circle. Imagine that the little line segments are rigid rods that pivot at their endpoints. Note that the point labeled A' moves in a straight line.
You can also adjust the lengths of the rods on the left of the figure.
Finally, you can step through a proof that A' traces out a straight line using the "Next" button in the control panel on the right.
These nine points all lie on a circle, which is called the Nine-Point Circle!
Load the diagram Ninepoint.T into Geometer and note that the nine points do, in fact, seem to lie on a circle. Move the vertices of the triangle and check to see that they seem to remain on a circle. (The circle, of course, changes, but always one circle goes through all the points.) In fact, if one of the angles of the triangle is bigger than 90 degrees, some of the points will lie outside the triangle, but the circle still manages to hit all of them.
OK, to see a proof of this fact, press the "N" key once. The first step of the proof will appear. Read it, understand it, and if you like, play with the shape of the triangle until you are convinced. Then press the "N" key again to get to the next step, and so on. To back up a step, use the "P" key, and to restart the proof from the beginning, press the "Start" button in the Geometer menu on the right of the drawing area.
The diagram Fagnano.T steps you though a proof of this fact using the "Next" button (or "N" key). At any stage in the proof, you can modify the shape of the original triangle and the shape of a test triangle which, if it is not the orthic triangle, will have a larger perimeter. In addition, the perimeters of the orthic triangle and the test triangle are continuously presented.
After you understand the proof, be sure to step through it again when the original triangle is not acute; when it has an angle larger than 90 degrees. See why the proof fails in this case.
Everyone is familiar with the elementary triangle centers, including the incenter (the intersection of the three angle bisectors), the orthocenter (the intersection of the three altitudes), the circumcenter (the center of the circumscribed circle), and the centroid (the center of mass of a triangle), but there are many others.
For example, if you find the points of tangency of the inscribed circle with the sides of a triangle and connect each to the opposite vertex, those three connecting lines will always meet at a single point known as the Gergonne Point.
Descriptions of many other such points can be found on Clark Kimberling's wonderful page: Triangle Centers.
Suppose you have found such a point and wonder if it is the same as any of the well-know centers. The triangle center finder will help, since it includes about 20 common centers.
Here's how to use it:
Load a copy of the file Finder.T and construct your point on the triangle that appears. Presumably there will be some construction lines, so make all of them invisible, and leave only your center visible. Change its color to something non-white, cyan, say.
Edit the text version of the Geometer file, find your point (which should be very near the end of the file), and assuming you used the color cyan, you should find in the description of your point something like this: [.cyan, .in] This means that on layer zero your point is colored cyan, and on all the others, it is invisible. Edit the [.cyan, .in] to [.cyan] and save the edited file. Now your point is visible on all layers.
Next, start pressing the "N" key on the keyboard, and different collections of triangle centers appear. If one seems to match your center, try moving the triangle vertices to see if it continues to match. If so, you may have found a well-known center. If not, continue pressing the "N" key until you find it or until all the possibilities are exhausted.
If you want to use the finder file again, don't save it after use; when you quit and Geometer asks if you want to save it, click on the "No" button.
Geometer Home Page
Email Tom Davis: firstname.lastname@example.org.