\section*{Problem 5 - 021235}

Given a line segment \(A B\) and a point \(M\) on it.

Construct squares \(A M D E\) and \(M B G H\) on the same side of the line segment \(A B\)! The centers of the two squares are \(R\) and \(S\).

What is the geometric locus of the midpoints of the segment \(R S\) ?
}

The segment \(AB\) is placed in a two-dimensional Cartesian coordinate system such that the following points have the following coordinates: \(A(0,0), B(a, 0), M(m, 0)\), with \(0 \leq m \leq a\).

According to the conditions of the problem, the points \(D, E, G\) and \(H\) then have the following coordinates: \(D(m, m), E(0, m), G(a, a-m), H(m, a-m)\).

The midpoints \(R\) of the square \(MADE\) and \(S\) of the square \(MBGH\) thus have the coordinates:

\[
R\left(\frac{m}{2}, \frac{m}{2}\right) \quad ; \quad S\left(\frac{a+m}{2}, \frac{a-m}{2}\right)
\]

The coordinates of the midpoint \(N\) of the segment \(RS\) in the two-dimensional Cartesian coordinate system are therefore

\[
N\left(\frac{a+2 m}{4}, \frac{a}{4}\right)
\]

Since the point \(M\) can be chosen arbitrarily on the segment \(AB\), i.e., \(0 \leq m \leq a\), the geometric locus of the midpoints of the segments \(RS\) is a segment \(KL\) parallel to \(AB\) with a length of \(\frac{a}{2}\), where its endpoints \(K\) and \(L\) have the following coordinates:

\[
K\left(\frac{a}{4}, \frac{a}{4}\right) \quad ; \quad L\left(\frac{3 a}{4}, \frac{a}{4}\right)
\]