(Level VI, part II)Geometricum geometriorum (HP III, Prisoner of "the circle")

Geometry Level 5

The perimeter of a circumference of radius \(5\) units and center \(O\) is divided into \(n\) equal parts. Find, as \(n\) tends to infinitum, the limit of the arithmetic mean of the lengths of all circumferences tangent to the first one, passing through the points of division and a fixed point \(B\) that is \(4\) units to \(O\)

Try Part I

Bonus: Generalise this problem when the radius \(5\) is \(a\) and \(4\) is \(b\) with \(0 \leq b < a\). The answer has a closed form, submit your answer to 3 decimal places.

Note: sorry, the picture is clearly not very good, but I hope it helps to understand better the problem.


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