A variable is created as a result of a measurement or a calculation. A newly created variable has a name and a definition. For instance, the variable myDistance in the example below is defined as the distance between points P and W in figure MyCube:

myDistance = |PW| : MyCube

In another example, variable myArea is a result of a calculation upon previously defined variables myVolume and myDistance:

myArea = 2 * myVolume / myDistance

Newly created variables are automatically recorded, along with their definitions, on the notepad.

Variable names can contain alphanumeric characters alone. Special and accented characters are not allowed. A variable name cannot contain spaces.

Variable names are case sensitive. Thus, myArea and MyArea will reference two distinct variables.

PI is reserved for the corresponding constant and cannot be used for newly created variables. However, it can be used in calculations.

Each variable has a numeric value. However, in most cases, this value remains transparent to the user. The user can create and solve Geometria problems without ever knowing the underlying values of the variables he or she has defined. To see the numerical value of a variable, type the variable in the calculator and click the Evaluate button. Due to roundup errors, the numeric values may slightly vary over the process of creating or solving the problem.

The value of a variable may change as a result of certain actions upon figures. If, in the example above, MyCube is scaled by the factor of 3, the value of myDistance will be implicitly multiplied by 3. The values of any variables that depend on myDistance, such as myArea, will be automatically updated as well.

However, the numerical value of the answer will be unaffected by actions upon figures. If, in the example above, the problem creator enters myDistance as the problem answer before scaling the figure, the value of the answer will not change, even though myDistance will carry a new value in all subsequent calculations. This note applies to all types of answer.

Therefore, the problem creator should consider (re)entering the answer before making the problem available to solvers.

After certain actions, the definition of a variable may become obsolete. For example, if point P or point W is erased, the definition of the variable myDistance above will become obsolete. Same if figure MyCube is removed. Such a variable is called pinned.

A pinned variable is no longer dependent upon its original definition and its value will no longer be updated. For example, erasing point P in figure MyCube will pin variable myDistance. If MyCube is subsequently scaled as described above, the value of myDistance will nevertheless stay the same.

If the action that pinned a variable is undone, the definition of the variable is restored and the variable becomes again targeted for updates.

Once a variable is defined, it can be used in calculations and referenced in actions, such as measuring, drawing and transforming figures. A variable can also be used to answer a problem. A pinned variable can still be used in calculations and references.

A variable exists as long as it is recorded on the notepad. Clearing the notepad is the only way to remove a variable. Use caution since this action cannot be undone.

When the document, such as a problem or a solution, is saved, all the variables recorded on the notepad are saved, too.

To rename a variable, select Rename Variable from the Edit menu. Alternatively, select the notepad record that defines the variable, right click and select Rename Variable from the popup menu.

A dialog will pop up. Select the variable you wish to rename from the drop-down list, enter the new name for the variable and click OK. The new name must be a valid variable name.

Click on the record that defines the variable on the notepad ahead of time, to have it automatically selected in the input box.

Renaming a variable will result in some of the notepad records being updated accordingly.

This action can be undone.

See also:

Calculations are performed upon valid expressions. Expressions can contain numbers, variables, the constant PI, operators, functions and parentheses.

The calculator operates in the context of the currently open document.

To calculate an expression myExpression and assign the result to a new variable myVariable, type in:

myVariable = myExpression

and click Evaluate. The variable name must be unique in the context of the currently open document. A new record will be added to the notepad. The newly created variable can be used in subsequent calculations and references.

If no document is open, the numeric result of the calculation will be displayed. No new variable will be created.

The following operators are supported:

The following functions are supported:

Arguments to functions are taken into parentheses.

Calculate the side of a cube whose volume is myVolume:

mySide = myVolume ^ (1 / 3)

Calculate the side adjacent to angle myAngle in a right triangle whose hypothenuse is myHypothenuse:

mySide = myHypothenuse * cos(myAngle)

To make the characters larger or smaller, select Font from the Options menu, then select the desired font size.

To switch to a different language, select Language from the Options menu, then select the desired language.

The application may behave erratically if the figures contain excessively sized or oddly shaped elements. Here are some guidelines for keeping the figures safe:

Problems, solutions and figures created with Geometria versions prior to 3.0 are no longer supported. However, we plan to have all future versions of Geometria backward compatible with 3.0.

Solution files created with Geometria versions prior to 3.1 must be opened using the same language in which they were saved.

To define the answer to the currently open problem, select Answer from the Edit menu. Select the answer type:

Open the Value pane.

The solver concludes the solution by providing a correct answer. Note, again, that the solver's answer is considered correct if it matches the creator's.

Have the solution opened. Select Answer from the Edit menu. Enter a valid expression or comma-separated labels, as appropriate. To minimize the input, have a notepad record and/or some elements, such as points, segments or a face, selected ahead of time in the selected figure. Click OK.

An evaluation of the answer (corect or incorrect) will pop up.

If the answer is correct, it will conclude the solution log. No more actions will be possible unless the answer is undone. You may want to save the solution to a file. A saved solution can later be played back.

See also:

As the solver progresses through the solution, his or her actions are recorded in the solution log. A saved solution can later be opened and played back.

To start playing back the solution, click the button.

To play back the next step, click the button.

To jump forward or backward, select an arbitrary record in the log.

To erase all records from the log, click the button. Use caution since this action cannot be undone.

To erase all records following the selected record, click the button. Use caution since this action cannot be undone.

To stop playing back the solution, click the button.

While the solution is being played back, it is freezed. To modify the solution, stop the play back first.

To comment on a solution step, right click on the corresponding record and select Comment from the popup menu. A dialog will be displayed. Type in your comments and click OK.

To view the comments, pause the mouse over the corresponding record. Commented records are highlighted in the solution log.

Geometria employs a fixed right-hand reference system. The Z axis of the reference system is parallel to the vertical edge of the screen. No knowledge of the reference system is necessary to manipulate figures, create and solve problems.

At any given time, the figure has a certain attitude. The attitude can be described in terms of angles the figure forms with the reference system. When a figure is created, it has the default attitude. The figure's attitude changes as the figure is rotated.

To revert a figure to the initial attitude, select Initial Attitude from the View menu or right click and select Initial Attitude from the popup menu.

When a figure is saved, its current attitude becomes its initial attitude. To revert to the attitude the figure had when it was created, before any subsequent rotations, select Default Attitude from the View menu.

See also:

Toggle the selector off and stroke a figure with the mouse to rotate the figure. The rotation speed and angle will depend on the stroke.

To pin a rotating figure, click on it.

See also:

To zoom in on a figure, select Zoom In from the View menu or click the button on the tool bar.

To zoom out, select Zoom Out from the View menu or click the button on the tool bar.

To resize a figure so it fits into the window, select Fit To View from the View menu or click the button on the tool bar.

See also:

To measure the distance between two points, proceed, alternatively, as follows:

The two points must belong to the same face.

A dialog will pop up. Fill in, if necessary, the points, assign a unique variable and click OK. A record will appear on the notepad, as below:

myDistance = |AB| : MyCube

where myDistance is the variable assigned to the distance between points A and B in figure MyCube. If MyCube is currently the only figure, the record will read:

myDistance = |AB|

This action can be undone.

See also:

To measure an angle, proceed, alternatively, as follows:

The segments that make the angle must belong to the same face and not be collinear.

A dialog will pop up. Fill in, if necessary, the points, assign a unique variable and click OK. A record will appear on the notepad, as below:

myAngle = <ABC : MyPyramid

where myAngle is the variable assigned to the angle between AB and AC in figure MyPyramid. If MyPyramid is currently the only figure, the record will read:

myAngle = <ABC

All angles are expressed in radians.

This action can be undone.

See also:

To measure the area of a face proceed, alternatively, as follows:

A dialog will pop up. Select, if necessary, the face from the list, assign a unique variable and click OK. A record will appear on the notepad, as below:

myArea = area[ABC] : MyCube

where myArea is the variable assigned to the area of the face that contains vertices A, B, C in figure MyCube. If MyCube is currently the only figure, the record will read:

myArea = area[ABC]

This action can be undone.

See also:

To measure the volume of a figure proceed, alternatively, as follows:

A dialog will pop up. Assign a unique variable and click OK. A record will appear on the notepad, as below:

myVolume = volume : MyCube

where myVolume is the variable assigned to the volume of figure MyCube. If MyCube is currently the only figure, the record will read:

myVolume = volume

This action can be undone.

See also:

To join two points with a segment, proceed, alternatively, as follows:

The two points must belong to the same face.

If a dialog pops up, fill in the points and click OK.

This action can be undone.

See also:

To draw a perpendicular through a point to a segment, proceed, alternatively, as follows:

The point and the segment must belong to the same face. The point may or may not belong to the segment.

If a dialog pops up, fill in the points and click OK.

If the point and the segment belong to two different faces at the same time, a second dialog may pop up. Select a face from the list and click OK.

This action can be undone.

See also:

To divide a segment in a given ratio, proceed, alternatively, as follows:

A dialog will pop up. Fill in, if necessary, the points and valid expressions for the numerator and denominator of the ratio. Click OK.

The ratio is defined as AX : XB, where X is the division point, AB the segment. For example, if the ratio is 1 : 2, X will be twice closer to A than to B. If the ratio is 3 : 1, X will be three times closer to B than to A.

The midpoint divides the segment in ratio 1 : 1 and therefore can be drawn as shown above. However, a convenience Draw Midpoint action offers a shortcut.

This action can be undone.

See also:

To draw the midpoint of a segment, proceed, alternatively, as follows:

A dialog may pop up. Fill in, if necessary, the points. Click OK.

Note that drawing the midpoint is equivalent to dividing the segment in ratio 1 : 1.

This action can be undone.

See also:

To divide an angle in a given ratio, proceed, alternatively, as follows:

The segments that make the angle must belong to the same face and not be collinear.

A dialog will pop up. Fill in, if necessary, the points and valid expressions for the numerator and denominator of the ratio. Click OK.

The ratio is defined as <AOX : <XOB, where OX is the division segment, <AOB the angle. For example, if the ratio is 1 : 2, OX will make with OB a twice greater angle than with OA. If the ratio is 3 : 1, OX will make with OA three times greater an angle than with OB.

The bisector divides the angle in ratio 1 : 1, therefore can be drawn as shown above. However, the convenience Draw Bisector action offers a shortcut.

This action can be undone.

See also:

To draw the bisector of an angle, proceed, alternatively, as follows:

The segments that make the angle must belong to the same face and not be collinear.

A dialog may pop up. Fill in, if necessary, the points. Click OK.

Note that drawing the bisector is equivalent to dividing the angle in ratio 1 : 1.

This action can be undone.

See also:

To draw the intersection point of the lines containing two given segments, proceed, alternatively, as follows:

Both segments, as well as the intersection point of their lines, must belong to the same face.

A dialog may pop up. Fill in, if necessary, the points. Click OK.

This action can be undone.

See also:

To draw a point on a given segment at a given distance from a given point, proceed, alternatively, as follows:

The point and the segment must belong to the same face. The point may or may not belong to the segment.

A dialog will pop up. Fill in, if necessary, the points and a valid expression for the distance. Click OK. Select a notepad record ahead of time to have it copied automatically to the distance input box.

This action may result in the construction of one or two distinct points.

This action can be undone.

See also:

To draw a segment from the end point of another segment so that the two segments make a given angle, proceed, alternatively, as follows:

A dialog will pop up. Fill in, if necessary, the points and a valid expression for the angle. Click OK. Select a notepad record ahead of time to have it copied automatically to the angle input box.

The angle must be greater than 0 and less than 180 degrees and be entered in radians, as opposed to degrees. For example, to lay an angle of 60 degrees, enter PI/3.

If the segment belongs to two different faces at the same time, a second dialog may pop up. Select a face from the list and click OK.

This action may result in the construction of one or two distinct segments.

This action can be undone.

See also:

The problem creator can rename a point in a figure by selecting Rename Point from the Draw menu.

A dialog will pop up. Enter the old and the new labels for the point in question and click OK. A valid label consists of a capital letter, which may be followed by one or more digits.

Select a point ahead of time to have it automatically copied into the input box.

Renaming a point may result in some of the notepad records being updated accordingly.

This action can be undone.

See also:

To scale a figure along a given segment by a given positive factor, proceed, alternatively, as follows:

A dialog will pop up. Fill in, if necessary, the points and a valid expression for the factor. Click OK. Select a notepad record ahead of time to have it automatically copied to the factor input box.

As a result of scaling, the previously measured distances, angles, areas and volumes will be recalculated and the assigned variables updated. See Variables for more details.

This action can be undone.

The solver cannot scale a figure that is part of the problem. However, the figures that have been created in the process of solving the problem may be scaled.

See also:

Shearing refers to an affine transformation where planes parallel to a fixed reference plane slide along a fixed line contained in the reference plane. The term affine implies that the transformation preserves the collinearity of points.

If XY is the fixed line in the reference plane, the shearing transformation is uniquely identified by points X, Y and the image of any point Z such that XZ is perpendicular to the reference plane.

To shear a figure, proceed, alternatively, as follows:

A dialog will pop up. Fill in, if necessary, the reference points (X, Y at the bottom and the image of point Z at the top). Click OK.

As a result of shearing, the previously measured distances and angles will be recalculated and the assigned variables updated. See Variables for more details.

The solver cannot shear a figure that is part of the problem. However, the figures that have been created in the process of solving the problem may be sheared.

This action can be undone.

See also:

To cut a figure with a plane, proceed, alternatively, as follows:

The selected elements must uniquely identify a plane. The plane must pass through the figure's interior.

If a dialog pops up, fill in the points and click OK.

The two resulting halves will be displayed in separate views. The original figure will be kept intact in its own view.

This action can be undone.

See also:

Two figures can be joined by either selecting Join Figures from the Transform menu or by clicking the button on the tool bar.

A dialog will pop up. Select the figures to join and click OK. A figure can be joined with a copy of itself. Therefore the same figure can be selected in both lists.

If the figures can be joined in one way only, a new view will appear with the combined figure in it. The two original figures will be kept intact in their own views.

If, however, the figures can be joined in more than one way, another dialog will be displayed asking if the figures should be joined in any of the possible ways. Click No to proceed with a more accurate definition of the joining elements.

Click Yes to have the figures joined in one of the possible ways. A preview will be displayed along with another dialog asking if the result should be kept. Click Yes to keep the result and conclude the action. Click No to get a preview of the next possible way of joining the figures.

It is recommended that, prior to joining the figures, the corresponding points and/or faces be selected. That will reduce to a minimum the subsequent dialog input. Note that faces can only be selected in the opaque view mode.

The joining faces must have the same shape, that is have equal angles and proportional sides at corresponding vertices. The joining faces do not have to have the same size. If need be, one of the figures will be automatically scaled to fit the other.

The action will be cancelled if the joint figure is not convex.

This action can be undone.

See also: