This chapter of the reference manual lists all the most common commands for Geometer. All of them are accessible using the GUI (graphical user interface). (Of course all of them are also accessible via the text editor.)
There is also an Advanced Reference Manual available.
Geometer is basically a simple constraint solving system. Each new item that appears in a Geometer diagram must be either an unconstrained element (like a free point), or it must be based on previously defined objects. The only geometry that's completely free is the free point, so most diagrams will be based on one or more of those.
Other unconstrained objects that can appear include text and numbers, but nothing depends on text, and numbers will be discussed in the advanced section.
In addition, there are some partially constrained points---a point that can move, but is constrained to lie on a line, a circle or a conic section, for example.
If the theorem you're trying to illustrate requires an unconstrained line or circle, just construct that line or circle from free points in one of the usual ways, and move or reshape the line or circle by moving the free points. (There are other ways to do this, but again, that's an advanced topic.)
For every redisplay of the current diagram, Geometer re-evaluates the entire file one line at a time, in order. (Actually, it's quite a bit cleverer than that, but the net result will always be the same as if it had done exactly this---except that the technique Geometer uses will get a redisplay a lot faster.)
This section is a description of all the commands you can use to enter geometric primitives. Of these, the most commonly used commands can be found as buttons in the upper part of the command menu on the right side of your window, but there are many more, all of which are listed in the pulldown menus under Primitives. In every case, if you click on the command's button, or use the pulldown menu, Geometer will produce a single instance of the primitive. If you hold down the Ctrl key when using the command button or pulldown, or double-click on the command button, Geometer will go into repeat mode for that command and you can make as many instances of it as you like. To exit repeat mode, use the Cancel Repeat Mode button at the bottom of the command menu (or type Ctrl-g), or click on another command without using the Ctrl button.
As soon as a new item is created, it is automatically selected. This is because the property-setting commands (like color) affect the selected item, and it's very common to create an item and decide immediately when you see it that you want it to be a different color or have a different style. Since newly-created items are selected, you merely need to click on the correct new color button to change the color of a new point, or click on the Point Type button to change how it looks on the screen.
This is not a list of all the constraints---there are additional constraints based on non-geometric primitives, but they are discussed in the advanced sections. All the commands below are accessed with the GUI (graphical user interface); the other constraints must be entered with a text editor.
Here are the ways to make new points from the graphical interface. Remember that many of them appear only in the Primitives pull-down menu. When new points are created, they automatically have a name generated, starting with "A", then "B", and so on. There's a command in the Edit pull-down menu to toggle this on and off. The command is called Point Names, and the keyboard shortcut is Ctrl-t.
Polygons are not particularly useful, except that you can find their area. The polygon's edges are not lines that can be used to create new primitives, for example. (You can, of course, add lines around the polygon's perimeter if you need to treat it both as a polygon and as a collection of surrounding lines.)
Arcs are not circles and you can't find the intersection of an arc with a line. Usually, arcs are simply used to make snazzy drawings, and the underlying circles are also in the diagram, but drawn in an invisible color so they don't clutter the view.
Whenever you change a property, you change that property for the selected geometric object and make that property the default for the next items you create (with the exception of the "invisible" color---if the current color is set to "invisible", new items are created in white). So if you want to make three red points and then 3 yellow points, set the color to red, create your three points, click somewhere else so that the last point you created is no longer selected, set the color to yellow, and make the final three points. Alternatively, you can enter the repeated creation mode by double-clicking (or by holding down the Ctrl key when you click) on the "create point" command, click locations for three points (which will be red), then click another location making a red (but selected) point, click on the "yellow" color command (which will change the red point to yellow, but will leave you in the repeat-creation mode), and then create the final two points.
If you want to change some property of an object you created a while ago, click on the object to select it and then click on the new color, line style, or whatever. Sometimes it is difficult to select the object you want because there are a bunch of other nearby objects that can't be moved away (angles are notoriously bad), but you can cycle through the different selection possibilities by simply holding down the Ctrl key while clicking again in the same place that is over multiple selectable items.
As was the case with the geometry creation commands, the most useful of the properties can be changed directly using buttons on the command menu, but there are a few others that are less common, and all of them except color are available under the Styles pulldown menu entry. The item's name, which is a sort of property, is also not available from the menu---the name can only be accessed via the editor or the dialog window you get when you type Ctrl-n.
Let's start with the color.
Every primitive has a color. The built-in standard colors include white, red, green, blue, yellow, magenta, and cyan. The "Color" button in the middle of the command menu shows the current default color and you can change it by clicking down on it and sliding the cursor up and down while the mouse button is down.
In addition to the colors mentioned above, Geometer supports a few special colors: invisible, smear, blink1, blink2 and blink3. When an object is invisible, you can't normally see it or select it. This is great for hiding auxiliary lines you used in a construction. If you need to work with invisible items, click on the Show Invis button in the command menu, and all the invisible items (and the non-invisible ones as well) will be visible, selectable, and movable, if they are points that are not completely constrained. Since one of the most common color changes made in Geometer is to make construction lines and circles invisible, there is a keyboard shortcut to make the current selection invisible: Ctrl-i.
There are three different blinking colors. When an item is drawn in one of these colors, it blinks---either between black and one of three colors or between two more closely related colors if you have a good enough graphics card on your computer. Blinking colors are great for highlighting features of interest in a step of a proof or construction.
Finally, there's a smear color. When an object is the smear color, when you drag a point that item will smear itself over the screen as long as the mouse is down. This is great for showing how one point moves in relationship with another, or to show how various interesting curves can be generated based on geometric constraints.
Geometer also lets you define new colors and use them, but you have to use the text editor. See the advanced section for more information.
The "invisible" color is also special in that if the button indicates that you're making invisible things, you will still make white ones. This is very handy, because when you're constructing something that will ultimately have a bunch of hidden features, they're drawn in white so you can work with them, but when they're correct and you want to hide them, you need only click on the Color button to make them invisible. After all, the "invisible" choice is on top. Otherwise you've have to scroll the color choices each time to get invisible, then click to unselect the primitive, then scroll back to a visible color.
(If you change a selected item to have the invisible color, it remains selected. This is because if you accidentally make it invisible, it disappears, and if it was an accident, all you need to do is change the color again to bring it back into view.)
Geometer tries to place hash marks and line names at the middle of the line, but that may not make sense for certain lines. Usually Geometer's guess is pretty good, but if you really need something else, you can usually put a point on the line where you want the name, and name the point. Make the point of type "No Mark". Line names are slightly different from points in that Geometer tries to avoid having them drawn on top of the line, so if you really need one of these, you can put two points close together on the line where you want the name, make a line connecting them, and name that line. And make the points of type "No Mark" or invisible.
Each primitive has a width, and those that are drawn with lines make use of this property. A width of 1.0 is the default, but the width can be set with the menu to 0.5, 1.0, 2.0, 3.0, 4.0, and 6.0. All figures that are drawn with lines, including lines, cirlces, arcs, conics, polygons, and Béezier curves are drawn with a width that is this number times the normal width. Since screen resolutions may not make these changes visible, using different colors to represent different features is often a better strategy. But if your main goal is to produce PostScript figures for publication in a black and white format, different line widths can be useful, and since printer resolutions are typically much better than screen resolutions, this works well.
There is currently a bug in the PostScript code to draw conic sections with widths other than 1.0. All conics are drawn with width 1.0.
Under the File pulldown menu appear most of the standard commands you'd expect to find---New starts a new empty diagram, Open opens an existing diagram, ReOpen repeats the previous Open command. ReOpen is useful if you're editing a Geometer file with your favorite (non-Geometer) editor, and you'd like to take a look at the results in Geometer after you've written them out from the other editor.
Save and Save As save the current file, either using the current name, or using a name that you provide. Insert inserts the contents of another file into the current one. The internal names of the inserted items are "munged" so that it's unlikely they'll conflict with any of the names in the current file. After you do an Insert, editing will be a bit ugly because of the funny new names, but sometimes it's a very useful command.
Print produces a PostScript file and tries to print it using GhostScript, and Save EPS makes an encapsulated PostScript file of the current drawing in the current directory.
Quit quits but asks you if you want to save any modifications you may have made, and Quit-No Save doesn't bother to ask---it just throws away all your work.
Under the Edit command, the most important command is Edit Geometry which will be described in much greater detail in the next couple of chapters.
Edit Name lets you change the name of the currently selected primitive. A dialog box appears containing the old name which you can edit, delete, or replace. When you exit from the dialog box, the new name is applied to the primitive. Warning: Don't try to put the double-quote character (") in a primitive name. Right now Geometer just throws them away if you try. I may fix this someday.
Delete Geometry is used to delete the selected item. This may be impossible because other items depend on it. For example, if your diagram consists of two points and the line connecting them, and you select one of the points and try to delete it, Geometer will complain, and won't let you do it until the line is deleted. The backspace key is a keyboard shortcut for Delete Geometry.
It is often a bit difficult to tell exactly what depends on what by just looking at the diagram, especially if there are some advanced features in use that don't show up graphically. If you use the Edit Geometry command to look at the entire structure of the file, it's usually a lot easier to figure out what's going on.
Describe Geometry gives a short description of the currently selected primitive. It tells you what it is, and what other items it's constrained by.
The Flip Angle command that simply reverses the sense of an angle. It is extremely easy to specify an angle backwards (in other words, to get the reflex angle of the one you really wanted), and if you create one that goes the wrong way around, just type Ctrl-a (or use the menu command) to flip it. Since you just created the angle, it will be selected, so you don't even have to select it---just issue the command.
Display Value toggles on or off whether the size of an angle, polygon or segment is displayed on the screen. If in your diagram, two angles seem to be equal but you can't tell for sure, make Geometer angles of both of them, select each one, and use this command so that their value will be displayed on the screen. The other values that can be displayed are the length of a line segment and the area of a polygon. If you've given the angle,polygon or segment a name, that name will be displayed; otherwise, Geometer will display the value with the internal name. This command only applies to the selected segment, polygon or angle. The keyboard shortcut is Ctrl-Shift-T.
But assuming you're just looking at a proof, there are basically three things you'll want to do---go to the beginning of the proof (which is where it should be when you load it), step forward to the next stage of the proof, or go back if you need to look at a previous step.
The three commands that do this are Start Proof, Next Step, and Previous Step. On the buttons, the names are shortened to Start, Next, and Prev, but they do the same thing. Since going to the next step is the most common thing you'll do, it has a keyboard shortcut---just type the "n" character. Typing "p" is the shortcut for Prev
In fact, almost every interesting theorem is a result of something similar---if you have a diagram with certain constraints, other constraints are required to hold. For something like the concurrence of three medians, the result is obvious to the eye, but other relationships may not be---that two segments are equal, for example, or that four points happen to lie in a harmonic relationship.
Geometer has a mechanism to help you search for such relationships in a semi-automatic fashion. Here is how to use it:
It works as follows. When you begin the test, Geometer looks at a very large collection of possible relationships and makes a list of all that seem to be true of the diagram in its initial configuration. Then, as you adjust the size and shape of the diagram, Geometer checks and rechecks the list to see which relationships continue to hold. If a relationship fails, it is dropped from the list. Finally, when you finish the test, Geometer displays all the relationships that held throughout the test.
If a relationship "obviously" holds, Geometer does not bother to list it. For example, if points A, B, C, and D are drawn, and also lines AB, AC, and AD, it is "obvious" that those three lines intersect at a point (they are defined to do so, after all), so Geometer will not report on this coincidence. But in the example above where the three medians meet at a point, the coincidence is not at all obvious, so Geometer will report it.
As an exercise, try drawing a triangle and its three medians in Geometer, and then test the diagram as described above. It should report the coincidence of those three lines. Now, add to the same diagram the three altitudes of the triangle and be sure to have points at the bases of the altitudes. Run the test again, and you will get a large set of interesting relations. Among other things, Geometer will find that the three medians and the three feet of the altitudes all lie on a circle (this is the famous nine-point circle). It will find some other circles as well---do you see why?
Unless they lie in a line, three points always lie on a circle, so obviously it is pointless to indicate that, but if four points lie on a circle, that is interesting. In the example above, there are six points on a circle, so Geometer goes nuts. It finds every set of four points that work. For example, if the six points are A, B, C, D, E, and F, Geometer will find that the 15 combinations: ABCD, ABCE, ABCF, ABDE, ABDF, ..., CDEF lie on a circle. Before presenting the information to you, however, it condenses it, reporting only that A, B, C, D, E, and F lie on a single circle.
Geometer, of course, does not guarantee that these relations hold exactly in every case---it simply checks to see that they hold to within a certain numerical tolerance for every configuration that you try. It may miss some relationships as well.
At present, these are the relationships that are checked:
The reason that you need to adjust the diagram between the beginning and end of the test rather than having Geometer simply wiggle the points around a bit is that there may be other relations you want to maintain but that are difficult to express using Geometer's features. For example, you may be looking for a theorem that holds only for acute-angled triangles (like Fagnano's Theorem), or you may want to look at sets of circles that all intersect each other. If you simply let Geometer wiggle the points, the triangle could be wiggled to make a non-acute-angled triangle, or so that some pair of the circles don't intersect.
It is good to do at least a little wiggling yourself---by chance, you may have drawn your diagram so that some relation happens to hold for that particular configuration, but would fail immediately with only a tiny movement of one of your points. Geometer is a bit conservative about throwing out possible relations, particularly for equalities of ratios, so it is a good idea to wiggle a few points. Usually only a very tiny amount of movement is necessary to get rid of these chance relationships, however.
Ratio equalities are always listed last since there can be a lot of them.
Geometer only considers visible objects in visible colors. Thus you can construct figures with lines and change them to be an invisible color if you do not want those lines to be examined for possible relationships. This becomes important with complex figures, as there are often many relationships to consider, and if you can reduce that number, your job will be easier.
The commands under Help will dump you into an internet browser looking at a lot of web pages of documentation. Since you're reading this, you may already have figured out how it works.
The commands under Help give various sorts of help. Three of them display documentation: Documentation. Tutorial and Reference Manual. You can also view a series of Usage Tips that give general hints on the usage of various Geometer features, and in addition, there is a Tool Tips command that causes Geometer to draw a little window over any button in the command area that explains its use if you stop the mouse over it. The feature is normally on. If you want to turn it on, click on the toggle button in the tips display.
The Show Text button in the command menu turns on and off the display of text. Sometimes a figure is complicated and the text overwrites some of the lines, and it's just easier to see what's going on without the added clutter of the text.
The Rpt Set Color button in the command menu is a quick way to change a lot of items to the same color. Before you press it, set the color button to indicate the new color you want. Then press Rpt Set Color, and you will be in repeat mode, where every item you click on will have its color changed to the set color. Use the Cancel Repeat Mode button at the bottom of the command menu to get out of this mode.
Some diagrams have a built-in script. If you want to run the script (and a well-designed diagram will tell you that a script exists), just press the Run Script button and the script will do its thing. The keyboard shortcut is S. Building a script is an advanced topic.
Finally, there may be a bunch of "Layer Control" buttons which are described in detail here. (They may not be visible---their display is controlled by an option.) Layers control stepping through proofs and constructions and lots of other things. For now, it's safer not to mess with those attractive red buttons.