Mitya is going to fold a square sheet of paper $ABCD$. Mitya calls a fold beautiful if side $AB$ intersects side $CD$ and the four resulting right triangles are equal. Before this, Vanya chooses a random point $F$ on the sheet. Find the probability that Mitya can make a beautiful fold passing through point $F$.

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Let's unfold a beautiful fold (see the figure on the right). Suppose the diagonal $B D$ and the fold line $S T$ intersect at point $O$. Triangles $B S O$ and $D T O$ are congruent by side and two angles. Therefore, $B O = O D$, and thus $O$ is the center of the square. This means that the fold line $S T$ passes through the center of the square. It is also clear that the converse is true - if the fold line passes through the center of the square, then the fold will be beautiful.
![](https://cdn.mathpix.com/cropped/2024_05_08_22585010b2534c7ab9ebg-15.jpg?height=736&width=1456&top_left_y=1205&top_left_x=312)

Point $S$ can occupy any position between $B$ and $C$, and point $T$ is located between $D$ and $A$. Therefore, to make a beautiful fold through point $F$, point $F$ must belong to triangle $B O C$ or triangle $A O D$.

The area of the figure bounded by these triangles is half the area of the square.

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