# G-0: Into the Plane

Welcome to the next phase of the course! In this module, we will learn how to use the GeoGebra software. It should be noted that although it is possible to create well drawn, accurate diagrams with this software, GeoGebra’s primary purpose is not to generate images. It is designed as a tool used to investigate mathematics.

The material this week has not been interspersed with exercises, but you will be asked frequently in the text to perform actions in GeoGebra. Therefore, you should have GeoGebra open when reading through the material. There will be four larger exercises to be submitted located at the end of lessons G-0, G-1, G-2, and G-3.

The focus of this lesson is to run GeoGebra and learn the basic functionality of the program.

## Starting GeoGebra

Similar to LaTeX, GeoGebra can either be accessed online, or downloaded and run on your computer locally. Go to http://www.geogebra.org and click “Downloads”.

Here you will see several different options. If you are installing the program on your machine, select the appropriate operating system under “GeoGebra Classic” and run the file when downloaded. GeoGebra can then be run locally at any time. This is the recommended method.

You may be directed to an alternate website instead of downloading the installer. This probably means that you do not have Java installed on your computer. GeoGebra is an application built off of Java, so you will need to follow the instructions on this website to download the Java software first.

GeoGebra can also be run directly in your browser. Clicking the plus icon at the top of the screen will allow you to start GeoGebra immediately. Note that the settings in your browser may be slightly different than what the default settings in the downloaded version are. Our instructions will assume students are using the downloaded version. It will be your responsibility to adjust settings and download completed work for submission if you use the online version.

If you choose to download and install GeoGebra on your computer, you may notice when you start the program that a Java window appears, explaining that it is downloading the application. This is normal, and it is simply GeoGebra updating itself to the most recent version. Mac users may be asked to save the application, and should always save the application in the same location every time to prevent multiple copies of the program appearing on their machines.

Let’s get started! Go ahead and start GeoGebra.

## Learning the Basics

### Tools

When GeoGebra starts, a new worksheet will open. There are four main areas that appear by default. The area on the right is the Graphics View. This is where all of the drawing happens. On the left, you will see the Algebra View, containing two categories: Free Objects and Dependent Objects. There is also an input bar at the bottom, where you can give commands for GeoGebra to perform, but we’ll touch on this in a later lesson.

The final area you will see in the window is the Toolbar appearing along the top of the window. Each square icon located at the top of the page represents a different tool, used to create or manipulate mathematical objects.

Let’s investigate. Select the **New Point** tool. Beside the Toolbar icons, you should see the following text appear:

**New Point**

Click on the Graphics View or on line, function, or curve.

This text gives the name of a tool (in bold) and instructions on how to use the tool. If you do not see this full message, you may want to increase the size of the GeoGebra window. Though it is not necessary to be able to see this message, it can give very useful information.

Now follow the instructions and click anywhere in the Graphics window. You have now created a point! You’ll note that this object has also been listed under Free Objects in the Algebra View.

Be aware that if you click again at a different location in the Graphics View, a second point will appear (try it). This occurs because the New Point tool is still the currently selected tool. In order to manipulate objects that you have created, you need to select the **Move** tool. This can be done by clicking on the tool directly, or using the Esc key. Once selected, objects can be move around the page by clicking on them and dragging. If you forget to switch to this tool and create an extra object that you did not mean to, simply use the Edit > Undo command, or delete the object.

Let’s now try using the **Line through Two Points** tool. Select this tool, then select points ‘A’ and ‘B’ on the screen. We now have a line passing through ‘A’ and ‘B’. You will note that this object appears in the Algebra View as well, but appears under Dependent Objects. Remember how we said that GeoGebra is not just a drawing program? Try moving ‘A’ and ‘B’ around to see what happens (don’t forget to switch to the Move tool).

As ‘A’ and ‘B’ are moved, the line follows, always passing through ‘A’ and ‘B’. This is because the line is defined in terms of ‘A’ and ‘B’. Though this is a very simple example, it illustrates a key idea behind GeoGebra: if we define one object ‘O’ (output) in terms of another object ‘I’ (input), then after any change to the object ‘I’, GeoGebra will automatically recompute and redisplay the updated version of ‘O’. Moreover, as we will see later, new objects defined in terms of ‘O’ will also be updated when this occurs.

### More New Points

Select the Line through Two Points tool again. This time, instead of clicking on two points already on the screen, click in any two locations in the Graphics View. We have now created three new objects: two points, and one line. In the algebra view, the two new points show up as free objects, and the new line shows up as a dependent object. There are two observations to take away from this.

The first is that if a tool requires a point to construct an object, a new point can be created for this purpose simply by clicking the Graphics View when using the tool.

The second observation is that some objects (such as lines) are designed to be dependent. That is, they need points in order to define how the object will appear.

### Variations on a Theme

You’ll notice that each tool icon has a small arrow in the bottom right corner. Clicking on this arrow will show a list of other tools you can use which are of a similar nature. For example, we note that the line tool we have used thus far always creates a line that continues to infinity. There may be cases where one wants to restrict a line to a segment. Locate and use the **Segment through Two Points** tool to create a segment in the Graphics View.

In other cases, we have that the same kind of object may have several different tools that can be used to create it in different ways. Lines are definitely such an object, but so are circles. Experiment with the three different tools used to create circles, and make sure you understand the differences:

**Circle with Center through Point**

Finally, there are tools that can take different types or numbers of objects as input and still generate a new object correctly. The **Midpoint or Center** tool is such a tool. Read the description on how to use it, and try the different methods suggested.

Hopefully this has given you a general understanding of how the tools in GeoGebra work. If you come across a tool that you are unsure of how to use, try reading the description written beside the Toolbar when the tool is selected to gain some insight. If you are still struggling, you can look up it’s description in the GeoGebra Manual online.

In addition to knowing where to look for information, it is also important to know what kinds of tools you have at your disposal when creating things in GeoGebra. We encourage you to experiment with any of the tools we have yet to discuss. In particular, you may want to try the tools below.

**Exercise** (nickname: `circumcircle`

)

Create a triangle ABC and a triangle DEF using line segments. Draw the circumcircle of triangle ABC (the circle passing through all 3 vertices) by drawing a circle through the three points, and label this circle “g”. Now construct the circumcircle of triangle DEF by doing the following: create two perpendicular bisectors on two different segments in the triangle. Find the point of intersection of these two lines, and label it G (feel free to hide the lines now). Then draw a circle with center G passing through D, and label it k.

Double-check that your construction is correct by dragging A, B, C, D, E, F around and seeing that the circumcircles still move with them.

**Note:** If you need help with changing and displaying labels and objects, see lesson G-1.