## The Chebyshev Walking Machine

The Chebyshev linkage is a mechanical linkage that converts rotational motion to approximate straight-line motion. It was invented by the 19th century mathematician Pafnuty Chebyshev who studied theoretical problems in kinematic mechanisms.

One of these linkages is shown above. It is called the Chebyshev walking machine. Notice that the mechanism seems to reproduce the legs of an animal. Click the Play button to start the animation.

## Five Squares to One

Move points A and B to resize the squares. Drag the slider to the extreme right to move the parts of the square. This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com
Snapshot

Contributed by: Izidor Hafner

## Dissecting an Octagon to Two Smaller Octagons

To dissect the octagon, move slider alpha to the extreme right and then move slider v to the extreme right.
This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com
Reference:

Contributed by: Izidor Hafner
Based on work by: Greg N. Frederickson

Snapshot

## Multiplication Table

Move the Column and Row sliders to choose the numbers you want to multiply.

## Cross to Isosceles Triangle Dissection

Move the Size slider to change the size of the cross and move the slider $\alpha$ to the extreme right to transform the figure.

The applet above is an example of a dissection puzzle. A dissection puzzle, also called a transformation puzzle tiling puzzle where a set of pieces can be assembled in different ways to produce two or more distinct geometric shapes (Wikipedia).

The cross to isosceles triangle dissection puzzle above was created by Henry Dudeney in 1897.

## Folding an Ellipse

Paper Folding Activity

1. Draw a circle and draw a point in its interior.
2. Mark a point anywhere on the circle.
3. Fold such that the two points coincide and crease well.
4. Repeat steps 2-3 as many times as you can.

What do you observer about the creases?Note: The green lines represents the folds on the circle.

## Polygon Names

Move the slider to change the number of sides of the polygon.   This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com
This applet is designed for elementary school pupils to help them memorize the names of polygons.

## The Inscribed Angle Theorem

In the figure below, angle BAC is an inscribed angle and angle BOC is the central angle of circle O. Both angles are subtended by the same arc. Make a conjecture about these two angles.
Check the Show/Hide Angle Measures check box and move any of the points. Is your conjecture always correct? Prove your conjecture.

## Circumcircle and Circumcenter

Click the check boxes and move the vertices of the triangle to explore. What do you observe?

In any triangle, a circle called circumcircle maybe drawn that passes through its vertices. The center of the circumcircle is called the circumcenter. The perpendicular bisector of the three sides of any triangle passes through its circumcenter.

## Napoleon’s Theorem

In the applet below, ABC is a triangle whose sides are also sides of equilateral triangles that contain the green line segments. Points P, Q, and R are centroids of the equilateral triangles.

1) What do you observe about the figure?
2) Move points A, B, and C. Are your observations still the same?

This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com

The Napoleon theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centers of those triangles themselves form an equilateral triangle.