On the side $B C$ of triangle $A B C$, a circle is constructed with $B C$ as its diameter, intersecting sides $A B$ and $A C$ at points $M$ and $N$.

Find the area of triangle $A M N$, if the area of triangle $A B C$ is $S$, and angle $A$ is $\alpha$.

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Triangle $M A N$ is similar to triangle $C A B$.

## Solution

CM and $B N$ are altitudes of triangle $A B C$. Triangle $M A N$ is similar to triangle $C A B$ with a similarity coefficient equal to $\cos \alpha$. Therefore, $S_{A M N}=S \cos ^{2} \alpha$.

![](https://cdn.mathpix.com/cropped/2024_05_06_14311972f49418f31424g-09.jpg?height=440&width=506&top_left_y=89&top_left_x=776)

## Answer

$S \cos ^{2} \alpha$.