# Dynamic Geometry

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.

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.