80. In a circle of radius $R$, two chords $A B$ and $A C$ are drawn. On $A B$ or its extension, a point $M$ is taken, the distance from which to the line $A C$ is equal to $|A C|$. Similarly, on $A C$ or its extension, a point $N$ is taken, the distance from which to the line $A B$ is equal to $|A B|$. Find $|M N|$.
80. If $\widehat{B A C}=\alpha$, then $|A M|=\frac{|A C|}{\sin \alpha},|A N|=\frac{|A B|}{\sin \alpha}$, i.e.

![](https://cdn.mathpix.com/cropped/2024_05_21_6282cf4a54d2a9633c48g-083.jpg?height=364&width=357&top_left_y=1362&top_left_x=162)

Fig. 17. $|A M|:|A N|=|A C|:|A B| ; \quad$ thus, $\triangle A M N$ is similar to $\triangle A B C$ with a similarity coefficient of $\frac{1}{\sin \alpha}$, therefore $|M N|=\frac{|B C|}{\sin \alpha}=2 R$.