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An Isoperimetric theorem

What you see in the applet below is three families of red and blue lines. The blue ones consist of smaller replicas of the red curve. What do you think of the lengths of the red and blue curves? In each family, which is longer?

(The small hollow circles are draggable. The red freehand curve is also modifiable by dragging any of its points. The number of break points could be changed by clicking on the number at the bottom of the applet.)

<hr> <h3> This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet. </h3> <hr>


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