1. **Stating the problem:** We need to find the size of the angle $\alpha$ in the given geometric configuration involving a rectangle, two circles (one smaller with radius $r$ and one larger with radius $R$), and various distances $I_1$, $I_2$, and $p$. 2. **Understanding the setup:** - The smaller circle of radius $r$ is inside the rectangle, centered near the bottom middle. - The larger circle of radius $R$ surrounds the smaller circle. - The angle $\alpha$ is formed at the bottom center beneath the smaller circle. - The distances $I_1$ and $I_2$ are vertical measurements from the bottom. - The distance $p$ is horizontal from the left edge to a diagonal line touching the larger circle. 3. **Key geometric relations:** - The angle $\alpha$ can be related to the tangent lines from the bottom point to the larger circle. - The tangent length and angle can be found using right triangle trigonometry. 4. **Formula for angle $\alpha$:** If we consider the point at the bottom center as vertex, and the tangent points on the larger circle define the angle $\alpha$, then: $$\tan\left(\frac{\alpha}{2}\right) = \frac{\text{opposite side}}{\text{adjacent side}}$$ Here, the opposite side can be the radius $R$ or a related vertical distance, and the adjacent side is related to $p$ or horizontal distances. 5. **Expressing $\alpha$ in terms of given parameters:** Assuming the tangent points create a right triangle with base $p$ and height $R$: $$\frac{\alpha}{2} = \arctan\left(\frac{R}{p}\right)$$ Therefore: $$\alpha = 2 \arctan\left(\frac{R}{p}\right)$$ 6. **Explanation:** - The angle $\alpha$ is twice the angle formed by the radius $R$ and the horizontal distance $p$. - This comes from the property that tangent lines from a point outside a circle form equal angles with the line connecting the point to the circle's center. 7. **Final answer:** $$\boxed{\alpha = 2 \arctan\left(\frac{R}{p}\right)}$$ This formula allows calculation of $\alpha$ if $R$ and $p$ are known.