12. Two tangent circles are inscribed in an angle of magnitude $\alpha$. Determine the ratio of the radius of the smaller circle to the radius of a third circle that is tangent to the first two and one of the sides of the angle.
12. Let (Fig. 3) \(O_{1}, O_{2}\) and \(O\) be the centers of the circles, \(M_{1}, M_{2}, M\) be the points of tangency with the side of the angle, \(r_{1}, r_{2}\) and \(r\) be the radii of the circles \((r_{1}<r_{2})\). By drawing a line through \(O_{1}\) parallel to \(M_{1} M_{2}\) until it intersects \(O_{2} M_{2}\), we obtain a right triangle with the hypotenuse \(r_{1}+r_{2}\), the leg \(r_{2}-r_{1}\), and the acute angle \(\alpha / 2\). By drawing a line through \(O\) parallel to \(M M_{2}\), we obtain two other right triangles.

![](https://cdn.mathpix.com/cropped/2024_05_21_6282cf4a54d2a9633c48g-066.jpg?height=281&width=496&top_left_y=206&top_left_x=607)

Fig. 3. 

Finding the segments \(\left|M_{1} M_{2}\right|, \left|M_{1} M\right|\) and \(\left|M M_{2}\right|\) from these three triangles, we write the equation \(\left|M_{1} M_{2}\right| = \left|M_{1} M\right| + \left|M M_{2}\right|\).

Answer: \(\left(\sqrt{\frac{1-\sin \frac{\alpha}{2}}{1+\sin \frac{\alpha}{2}}}+1\right)^{2}\).