Octagon to Square Dissection

The applet below shows the octagon to square dissection.

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:
"Bennett's Octagon-to-Square Dissection" from the Wolfram Demonstrations Project http://demonstrations.wolfram.com/BennettsOctagonToSquareDissection/ Contributed by: Izidor Hafner Based on work by: Greg N. Frederickson

Square Root of a Sum? - Geometric Interpretation

We know that for non-negative numbers  Can we apply the same for the sum? 
In the example below a and b are the legs of a right triangle.  According to the Pythagorean Theorem the hypotenuse is

• Drag to the left the blue point A to rotate the leg AC around point C until the two legs form one segment.
  
• Drag to the left the orange point A to rotate the hypotenuse around point B until it comes to the base of the triangle.
• Compare the lengths  

Irina Boyadzhiev, 6 April 2013, Created with GeoGebra

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Irina Boyadzhiev's  GeoGebra Applets

The Sum of the First n Natural Numbers - Dynamic Geometric Proof

This applet gives a dynamic proof of the formula for the sum of the first n natural numbers.


Irina Boyadzhiev, 6 March 2013, Created with GeoGebra

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• See the general case - Arithmetic Series.

Irina's GeoGebra Applets

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