4. A sphere is inscribed in a pyramid whose base is a rectangle with diagonal $d$. The lateral edges of the pyramid are inclined to the base plane at an angle $\beta$. Find the radius of the sphere.
Solution. Let $O$ be the center of the sphere circumscribed around the pyramid $S A B C D$ with base rectangle $A B C D$, let $R$ be the radius of the sphere, let $E$ be the intersection

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of the diagonals of rectangle $A B C D$, and let $M$ be the midpoint of segment $C S$. Then, triangle $S E C$ is a right triangle with acute angle $\beta=\angle S C E$, and point $O$ lies on line $S E$ and is equidistant from points $S$ and $C$, so $O M \perp C S$. Since $\angle S O M=\angle S C E=\beta$, as angles with mutually perpendicular sides, we have

$$
R=\overline{O S}=\frac{\overline{S M}}{\sin \beta}=\frac{1}{2} \frac{\overline{C E}}{\cos \beta \sin \beta}=\frac{d}{2 \sin 2 \beta} .
$$

## IV year