Constructing the Radius

We already know that the radius of the twin circles is  where a and b are the radii of the small semicircles in the arbelos. There is a nice way of constructing this radius without doing any computations. Let's investigate how this works!

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Drag the red and blue points in this construction to change a and b. The green segment has the length of our radius r.

Why?

Try to explain why the construction above really gives us the twin circles' radius. The following steps will help you.
  1. Look for similar triangles in the construction above and write them down.
    Hint:

  2. Use the similar triangles from (1) to write down two ratios that include r, a and b.
    Hint 1:
    Hint 2:

  3. Use the two equations from (2) to caclulate r.
    Hint:
  4. Bonus: can also be written as . Show that these two expressions are equal.
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