The Sum of Squares - dynamic proof

This applet gives a  dynamic proof of the formula for the sum of the squares of the first n natural numbers.  We start with three times the sum of the squares and rearrange the parts of one of the sums.

• Click on "Color the third column".
• Drag the slider to rearrange the parts of the third column.

The area of the formed rectangle equals three times the sum of the squares.

• Click on  Show the formulas  to see the result.

Irina Boyadzhiev, 13 February 2013, Created with GeoGebra

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Irina's GeoGebra Applets

Angle Bisector of a Right Triangle

Move point P.  What do you observe?

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

1. What can you say about the diagonal of the outer square in relation to its angles? in relation to the right triangles formed by the square? 
2. What can you say about the diagonal of the outer square in relation to the inner square?
3. What conjecture can you make from your observation in 2?

Two Squares and a Quadrilateral

A common vertex is shared by two squares A. Quadrilateral HIJK is formed using the segments joining the vertices and the midpoints of the diagonals of the square. What is the shape of the quadrilateral?

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

Questions

1. Move the vertices of the two squares. Are your observations still the same? 
2. Make a conjecture about your observation. 
3. Prove your conjecture. 

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Reference
 http://www.ies.co.jp/math/java/geo/twosqsq/twosqsq.html

van Aubel's Theorem

Squares are placed outwardly at each side of quadrilateral ABCD. The center of the squares on the opposite sides are connected with segments (PN and MO). Move points A, B, C and D and observe what happens. Make a conjecture about your observations.

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

Van Aubel's Theorem

Given an arbitrary planar quadrilateral, place a square outwardly on each side, and connect the centers of opposite squares. Then van Aubel's theorem states that the two lines are of equal length and cross at a right angle.

Click here to download GGB file.