Geometricus geometriorum (HP II, secret "semi-tyre")

Geometry Level 5

A semicircumference of radius \(\pi\) is divided into \(n + 1\) equal parts and is joined any point of the division with the ends of the semicircumference, forming a right-triangle with area \(A_{k}\) .

Find the limit as n tends to infinity of the arithmetic mean of the areas of these triangles.


Bonus: Generalise when the radius is \(r\)


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