10.023. Given a square, two of its vertices lie on a circle of radius $R$, and the other two lie on a tangent to this circle. Find the length of the diagonal of the square.
## Solution.

Let $x$ be the side of the square (Fig. 10.23), in $\triangle O E A ~ \angle O E A=90^{\circ}$, $E A^{2}=O A^{2}-O E^{2}, R^{2}-(x-R)^{2}=\frac{x^{2}}{4}, \frac{x^{2}}{4}+x^{2}=2 x R ; x=\frac{8 R}{5} ; A C=x \sqrt{2=}$ $=\frac{8 R}{5} \cdot \sqrt{2}=1.6 R \sqrt{2}$.

Answer: $1.6 R \sqrt{2}$.