(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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