阿基米德双子圆

http://www.geogebra.org/en/examples/frisbee/worksheets/twin_circles_radius.html

作图的方法这里的介绍更精彩http://www.math.utah.edu/mathcircle/10-17-2007-notes-hanson-arbelos-solutions.pdf

Radius of the Twin Circles

In order to construct the Archimedian Twin Circles we need to know their radius. Right now, we don't know yet if they really both have the same radius, so let's find out!

• Left Twin Circle
• Right Twin Circle
• Archimedes' Proof

Left Twin Circle

In the sketch below you see an arbelos with one of the twin circles. You can drag point N along the diameter to change the shape of the arbelos.

Your task is to calculate the twin circle's radius r in terms of the two small semicircles' radii a andb. Note: the large semicircle has center point M and its radius is R = a + b.

1. Use the Pythagorean theorem for the right triangles CEF and MEF. This lets you express the segment EF in two ways.
   Hint: 
   
2. Use the two equations from (1) to get one equation without segment EF.
   Hint: 
   
3. Let's now try to get a formula for r in terms of a and b. 
   Start with writing all segments used in (2) in terms of a, b, r and R.
   Hint 1: 
   Hint 2: 
   Hint 3: 
   
4. Now plug your results from (3) into the equation from (2) and solve it to get our twin circle's radius r.
   Hint 1: 
   Hint 2: 

Right Twin Circle

Now you know how to get the radius r of the left twin circle. Does the right twin circle have the same radius? Let's make sure and proof it! The construction below shows both twin circles. 

• Calculate the radius r of the right twin circle by using the same strategy as above. 
• Do you get the same result?
  

 

Archimedes' Proof

You would like to know how Archimedes proved that the twin circles have same size? Find out inProposition 5 of his Book of Lemmas.