On the plane, there is a unit-length segment $AB$ and a point $P$ outside it. The segment $AB$ is rotated around $P$, sweeping out a circular arc of area $t$. Determine the set of points around which, when $AB$ is rotated, the resulting circular arc also has an area of $t$.
Let's examine the area of the circular ring formed by rotating $AB$ around an arbitrary point $X$. If the reflection of $X$ with respect to the perpendicular bisector of $AB$ is $X'$, then by symmetry, the areas of the circular rings corresponding to $X$ and $X'$ are equal. Therefore, it is sufficient to examine the points for which $XA \leq XB$.

Let the orthogonal projection of $X$ onto the line $AB$ be $T$, and denote the distance $TB$ by $x$. If $T$ lies on the segment $AB$, then $x \leq 1$, and the area of the circular ring corresponding to $X$ is $t(x) = XB^2 \cdot \pi - XT^2 \cdot \pi = TB^2 \cdot \pi = x^2 \pi$ (Figure 1).

If $T$ lies outside the segment $AB$, then $x > 1$ and the area of the circular ring corresponding to $X$ (Figure 2):

$t(x) = XB^2 \cdot \pi - XA^2 \cdot \pi = (XT^2 + TB^2) \cdot \pi - (XT^2 + TA^2) \pi = (TB^2 - TA^2) \pi = (x^2 - (x-1)^2) \pi = (2x - 1) \pi$.

It can be seen that the area of the circular ring does not depend on $X$, but only on its orthogonal projection onto the line $AB$. The function describing the area of the circular ring is:

$$
t = \begin{cases} 
x^2, & \text{if } 0 \leq x \leq 1, \\
2x - 1, & \text{if } x > 1.
\end{cases}
$$

This function is monotonically increasing, so for each area of the circular ring, there is only one corresponding $x$. Therefore, the geometric locus of the points we are looking for - taking into account the symmetry mentioned in the introduction - is the perpendicular from $P$ to the line $AB$ and its reflection with respect to the perpendicular bisector of $AB$. If we consider the circle as a degenerate circular ring, then the intersection points of the two lines and $AB$ also belong to the set. If we do not consider the circle as a circular ring, then these two points should be excluded from the set.

Based on the work of Péter Frenkel (Fazekas M. Fóv. Gyak. Gimn., II. o.t.)

![](https://cdn.mathpix.com/cropped/2024_05_02_f1884f89e6aba940b295g-1.jpg?height=1028&width=1030&top_left_y=1096&top_left_x=537)

![](https://cdn.mathpix.com/cropped/2024_05_02_f1884f89e6aba940b295g-2.jpg?height=1169&width=1229&top_left_y=115&top_left_x=429)