# Sample Geometer Diagrams

This page contains descriptions of various Geometer diagrams together with downloadable files containing those diagrams. If you find any of the descriptions to be interesting, download the Geometer file and open it with Geometer.

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.

## Index to Sample Diagrams

A High Tech Spirograph (elementary, advanced)
The Area of a Circle (elementary)
Visual Trigonometric Functions (elementary)
Conic Section Passing Through Five Points (elementary)
Lissijous Figures (elementary)
Harmonic Numbers (elementary)
A Sum of Angles (intermediate)
Construction of External Tangents (intermediate)
The Euler Line (intermediate)
Construction of a Regular Heptadecagon (intermediate)
Poncelet's Theorem (intermediate)
The Nine-Point Circle (intermediate)
Fagnano's Problem (intermediate)
Brahmagupta's Theorem (intermediate)
Circle Tangents Problem (intermediate)

## Viewing the Diagrams

After you've downloaded a diagram, there are a few ways to view it. If you've done the Geometer installation, you can just double click on the file name and Geometer will run on that file automatically. If you just downloaded the executable, you can drag the file icon on top of the Geometer icon. Finally, you can just double-click on the Geometer icon and use the "Open" command in the "File" pull-down menu to find and load the downloaded Geometer diagram file. Files having a suffix of .T or .D are Geometer diagrams.

There are various things that can be done, depending on the diagram:

• You can drag some points of a diagram by clicking down on them with the right mouse button, dragging, and releasing them.
• You can sometimes use the "Next" and "Prev" buttons in the control panel on the right of the Geometer window to step through a proof or construction. The "N" and "P" keyboard keys are shortcuts for the "Next" and "Prev" buttons. You can return to the beginning of a proof or construction by pressing the "Start" button.
• Finally, if it is not grayed-out, you can press on the "Run Script" button to run a canned Geometer script. Only a few files have this feature.

## A High Tech Spirograph

Load the file Spirograph.T into Geometer and press the "Run Script" button. A nice figure will by traced out by a series of dots.

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); ```

to be

``` v1 = .f.rpn(28.000000); v2 = .f.rpn(7.000000); ```

Make the change and press "Run Script" again.

## The Area of a Circle

This Geometer file gives a nice illustration that the area of a circle is πr² (pi r-squared). Load the diagram circ.T into Geometer, look at the figure closely, and then click on the "Run Script" button in the control panel on the right.

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.

## Visual Trigonometric Functions

Here is a nice way to visualize how the standard trigonometric functions like sine, cosine, tangent, cosecant, et cetera, are geometrically related to the angle. Load the diagram TrigFuncs.T into Geometer, look at the figure closely, and then click on the "Run Script" button in the control panel on the right.

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.

## Conic Section Passing Through Five Points

This diagram illustrates the fact that for almost any five points on the plane, there is a unique conic section that passes through all of them. The conic sections include the circle, the ellipse, the hyperbola, and the parabola.

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.

## Lissijous Figures

Load the file Lissijous.T into Geometer, and press the "Run Script" button. A lissijous figure will be traced out with dots on the screen. What is happening is that two points are moving around two different circles at different speeds, with a horizontal line passing through one and a vertical line passing through the other. At the intersection of these two lines, a series of points is drawn, and this traces out a lissijous figure.

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.

## Harmonic Numbers

Load the file Harmonic.T into Geometer, and press the "Next" button (or "N" key) repeatedly to step through a simple geometric construction of points on a line spaced harmonically.

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.

## A Sum of Angles

This file steps you through a proof that a certain set of three angles adds to 90 degrees. Load the file into Geometer, read the problem statement, and then step through the proof by pressing the "Next" button. The "Prev" button backs up a step and the "Start" button goes back to the beginning of the proof. You can use the "N" and "P" buttons as shortcuts for "Next" and "Prev".

## Construction of External Tangents

This diagram leads you step by step through the construction of a pair of common external tangents to two circles. Use the "Next" and "Prev" buttons (or the "N" and "P" keys as shortcuts) to move forward or backward in the construction.

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?

## The Euler Line

This diagram steps through the proof that the center of mass of a triangle, its orthocenter, and its circumcenter all lie on a straight line called the Euler Line. Use the "Next" and "Prev" buttons (or the "N" and "P" keys as shortcuts) to move forward or backward in the proof. At one stage in the proof, you will be asked to press the "Run Script" button.

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.

## Construction of a Regular Heptadecagon

This diagram steps through the construction of a regular heptadecagon (a regular 17-sided figure). Simply use the "Next" and "Prev" buttons (or the "N" and "P" keys) to step forward and backward through the construction. The "Start" button goes back to the beginning of the construction.

## Poncelet's Theorem

This diagram illustrates but does not prove Poncelet's Theorem.

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.

For many years it was unknown whether it was possible to convert circular motion into perfect linear motion. The mechanical linkage discovered by Peaucellier proved that it can be done.

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.

## The Nine-Point Circle

I think this is one of the most miraculous results in elementary euclidean geometry. Take any triangle and find the three midpoints of the sides. Next, drop altitudes from each vertex to the opposite sides, and find the feet of those altitudes; the places where the altitudes meet the opposite sides. Finally, the three altitudes meet at a point called the orthocenter and find the three midpoints on the altitudes between the orthocenter and the vertices of the original triangle.

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.

## Fagnano's Problem

Fagnano's problem is to construct a triangle of minimum perimeter inside a given acute triangle (an acute triangle is one having all its angles less than 90 degrees). The solution is the orthic triangle that connects the feet of the altitudes of the original triangle.

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.

## Brahmagupta's Theorem

You may have heard of Brahmagupta's formula to determine the area of a cyclic quadrilateral, but this is different. It is a theorem concerning cyclic quadrilaterals. Use the "Next" key to step through a proof.

## Circle Tangents Problem

This diagram steps through a proof of a theorem concerning tangent lines to circles.

## Triangle Center Finder

Warning: this file is a bit tricky to use. Try it only after you're pretty comfortable with Geometer.

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.