[ Rectangles and squares. Properties and characteristics ] [ Congruent triangles. Criteria for congruence (other).]

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 congruent. Before this, Vanya chooses a random point $F$ on the sheet. Find the probability that Mitya can make a beautiful fold passing through point $F$.
Let's unfold the beautiful fold (see the figure on the right). Suppose the diagonal $BD$ and the fold line $ST$ intersect at point $O$. Triangles $BSO$ and $DTO$ are congruent by side and two angles. Therefore, $BO = OD$, and thus $O$ is the center of the square.

Thus, the fold line $ST$ 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_06_ccb5665c9ecd710ea159g-33.jpg?height=732&width=1456&top_left_y=111&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 $BOC$ or triangle $AOD$.

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

## Answer

$1 / 2$