Inscribed Circle Circumference Result

Given a circle, construct on a diameter, two smaller circles tangential to each other and the large circle as shown. Which would be larger? The circumference of the large circle or the sum of the circumferences of the two smaller circles? Is that always the case?

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Inscribed Circle Circumference Result

a) Can you explain (prove) why the result you observe is true? Can you make a conjecture (and prove) what'll happen if 3, 4 or more such circles are constructed on the diameter of a circle?

b) Suppose more and more smaller circles are inscribed until they become minutely small so that they virtually become 'points' on the diameter. Will the sum of the circumferences of these very small circles gradually approach the length of the diameter as they get smaller? Explain why, or why not!

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Michael de Villiers, 6 April 2010.