\section*{Problem 3 - 081043}

In a plane \(\epsilon\), let \(k\) be a circle with a given radius \(r\); furthermore, let a natural number \(n>2\) be given.

A diameter \(P Q\) of \(k\) is divided into \(n\) equal parts; the division points are \(R_{1}, R_{2}, \ldots, R_{n-1}\), such that

\[
P R_{1}=R_{1} R_{2}=\ldots=R_{n-2} R_{n-1}=R_{n-1} Q
\]

holds.

One of the two half-planes into which \(\epsilon\) is divided by the line \(g_{P Q}\) is called \(H\), the other \(H^{\prime}\). Then let \(c_{i}\) be the semicircle in \(H\) over \(P R_{i}\), and \(c_{i}^{\prime}\) the semicircle in \(H^{\prime}\) over \(R_{i} Q\), and finally \(b_{i}\) the curve composed of \(c_{i}\) and \(c_{i}^{\prime}\) \((i=1,2,3, \ldots, n-1)\).

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1267.jpg?height=671&width=805&top_left_y=795&top_left_x=634)

Calculate the areas of the regions into which the circular area is divided by any two adjacent curves \(b_{1}, \ldots, b_{n-1}\) or by \(b_{1}\) or \(b_{n-1}\) and the respective semicircle!
}

Defining additionally \(R_{0}:=P, R_{n}:=Q\), and \(b_{0}\) and \(b_{n}\) as semicircles over the diameter \(P Q\) in \(H^{\prime}\) and \(H\) respectively, we are now generally asked for the area between the two curves \(b_{i-1}\) and \(b_{i}\) with \(1 \leq i \leq n\). Let this area be denoted by \(F_{i}\).

The area \(F_{i}\) can be decomposed into the part that lies in \(H\) and the part that lies in \(H^{\prime}\). The first part can be represented as the difference of the semicircles in \(H\) over the diameters \(P R_{i}\) and \(P R_{i-1}\), and the second part as the difference of the semicircles in \(H^{\prime}\) over the diameters \(R_{i-1} Q\) and \(R_{i} Q\).

Thus, the area of the figure \(F_{i}\) is calculated as

\[
\frac{\pi}{8} \cdot\left(\frac{i^{2}}{n^{2}}-\frac{(i-1)^{2}}{n^{2}}+\frac{(n-(i-1))^{2}}{n^{2}}-\frac{(n-i)^{2}}{n^{2}}\right) \cdot(2 r)^{2}=\frac{\pi \cdot r^{2}}{2 \cdot n^{2}} \cdot(2 i-1+2(n-i)+1)=\frac{1}{n} \cdot \pi r^{2}
\]

The circle is thus divided into \(n\) equal-area segments.