262. Inscribe the largest square in the given semicircle.
262. If the side of the sought square is $x$, and the radius of the given semicircle is $r$, then

$$
r^{2}=x^{2}+\frac{x^{2}}{4}=\frac{5}{4} x^{2}
$$

or

$$
\frac{4 r^{2}}{5}=x^{2}, \text { i.e. }{ }_{5}^{4} r: x=x: r
$$

thus, the sought side can be constructed as the mean proportional between the radius and its four fifths.

Of course, this side can also be constructed based on purely geometric considerations, using the method of similarity. Let the square $F D E H$ be the sought one (Fig. 61). Draw the diagonal $D O$ in the rectangle $D F O K$ and from an arbitrary point $N$ on it, draw $N P \| A B$ and $N M \| D F$.

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-138.jpg?height=420&width=715&top_left_y=1229&top_left_x=956)

Fig. 61. The resulting rectangle $O P N M$ is similar to $D K O F$, since in both the base is equal to half the height. From this, the construction follows: construct a rectangle $O P N M$ where $P O=2 M O$, and connect $N$ with the center $O$. Extend $O N$ until it meets the circle at point $D$, and draw $D E \| A B$. $D E$ is the side of the sought square.