Consider the circle shown below:

The diameter BM is divided in to 11 equal length segments: BC,CD,DE,...LM. Then we complete semicircles with diameters CM,DM,...KM,LM on the top of BM and semicircles with diameters BC,BD,BE,....BL on the bottom of BM. These semicircles divide the circle in to 11 blade-shaped objects. Show that all blades have the same area!

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Name the 'blades' by the segment that each intercept on the diameter i.e B(xy).

By counter clockwise rotation clearly:

1. B(BD) = B(KM)

2. B(BC) = B(LM)

Subtracting 2 from 1 gives

3 B(BC)= B(CD)

In a like manner QED.

Let distance between 2 adjacent parts be 1. Let there be N slices. Let s = N-1-r

Area of the rth slice /pi = (r+1)^2 - r^2 + (s+1)^2 - s^2.

The RHS is just a function of N.