Dynamic geometry software allows for traditional constructions with compass and straightedge to be performed by computer and then to be manipulated in a dynamic way.

This page shows a few examples using *Geometer's SketchPad* and
*Cabri Geometry II*. At present, they are simple examples to
illustrate how the software works. In this case, a Java version of GSP
(*Java SketchPad*) and of Cabri (*CabriJava*) are used to
construct objects which can be manipulated by the user. At the bottom of
this page, three dimensional objects are shown, using *Cabri 3D* and examples of other uses of dynamic geometry are shown with *GeoGebra*.

The **Circle** example allows a user to manipulate a circle by
dragging either the centre or a point on the circumference. As these are
dragged, the circle changes size, and the radius, perimeter and area are
recorded automatically. The ratios of circumference to radius and area
to squared radius remain constant, however. Try the Circle applet for yourself.

The **Medians** of a triangle are constructed and seem to be
concurrent. As you move the vertices of the triangle around, the medians
are correspondingly changed, but they still appear to be concurrent,
suggesting that this is a general property of medians. Try the Medians applet for yourself.

The **Altitudes** of a triangle are constructed and seem to be
concurrent. As you move the vertices of the triangle around, the
altitudes (or perpendicular heights) are correspondingly changed, but
they still appear to be concurrent, suggesting that this is a general
property of altitudes. Try the Altitudes applet for yourself.

The **Perpendicular bisectors** of the sides of a triangle are
constructed and seem to be concurrent. As you move a vertex of the
triangle around, two perpendicular bisectors are correspondingly
changed, while the third is not changed. The three perpendicular
bisectors still appear to be concurrent, suggesting that this is a
general property of triangles. Try the Perpendicular bisectors applet for
yourself.

A pair of lines intersects at a point, and the **Angles** formed
by the lines are measured. The opposite angles have the same size,
regardless of the movement of the lines. The diagram helps us to see why
this must be the case ... since the pair of angles that comprise any
line must necessarily add to 180 degrees. Each of the opposite angles
must be the difference between 180 degrees and the other angle. Try the
Angles applet for yourself.

Two **Angles in a circle** are formed from points on the circle.
The diagram helps us to see the congruence of the angles, regardless of
the choices of points. Try the Angles in
a circle applet for yourself.

Two **Parallels** are intersected by a transversal line, resulting
in several angles. The applet allows you to see which angles are
congruent. Try the Parallels applet for
yourself. Here is a *GeoGebra* version.

When the midpoints of the sides of a **Quadrilateral** are joined,
another quadrilateral results. using the diagram helps to see what is
special about this new quadrilateral. Try the Quadrilateral applet for yourself.

The **Pythagorean Theorem** can be proved in many ways. This
diagram uses shears to show a version of Euclid's proof. The applet is a
slight variation on the one provided by the publishers of GSP. Try the
Pythagorean Theorem applet for yourself.

**Trig tracers** show the paths of points on a circle, leading to
sine and cosine curves. The applet is a slight variation on the one
provided by the publishers of GSP. Try the Trig tracers applet for yourself.

**Vertically opposite angles** are formed by a pair of
intersecting lines. By manipulating the lines, the sizes of the angles
can be compared. Try one of the applets, JSP version or GeoGebra version for yourself.

When an object is subjected to **Two reflections** , there are
various possibilities for the composite transformation. These can be
seen by manipulating the reflections as well as the original object. Try
the Two reflections applet for
yourself.

**Transformations**

Transformations in the plane can be studied using dynamic geometry. The examples here have been made with *GeoGebra* software.

The isometries are rigid transformations that preserve distances. The three most important isometries are those that reflect, translate or rotate the plane. Objects and their images are congruent.

Enlargements (or dilations) are not isometries, but are examples of similitudes. Images are similar to the original objects; that is, they have the same shape, but are not necessarily the same size.

**Three-dimensional geometry**

Three-dimensional objects can also be manipulated. The examples here have been made with the remarkable *Cabri 3D* software. If you
do not have the necessary (free) *Cabri 3D* plug-in on your browser, you
will be directed to download it. There is more information from the software developers here, as well as many more examples.

The five Platonic solids comprise all the polyhedra with faces that are congruent regular polygons and for which each vertex is the same. There are only five of these: the tetrahedron, the cube (hexahedron), octahedron, dodecahedron and icosahedron.

The dual of a cube is an octahedron, a surprising link between these two Platonic solids.

The volumes of pyramids and prisms are related, as can be seen with this pentagonal example.

**Other uses of dynamic geometry**

Dynamic geometry software can be used for purposes other than directly geometric ones. Here are some examples, using *GeoGebra*, a free software package available here.

The normal distribution is of critical importance in studying sampling distributions as well as other purposes. (Download the original *GeoGebra* file by right-clicking here.)

More coming.