92. Three consecutive vertices of a rhombus lie on the sides $A B, B C, C D$ of a given square with side 1. Find the area of the figure filled by the fourth vertices of such rhombi.
92. Answer: $S=1$.

Let $K, L, N$ be the vertices of the rhombus on the sides $A B, B C$, and $A D$ of the square (Fig. $52, a$). Note that the length $K B$ is equal to the distance from point $M$ to the line $A D$. Therefore, if we fix point $K$, the possible positions of point $M$ fill a certain segment $M_{1} M_{2}$ parallel to side $A D$.

The lower position $M_{1}$ of point $M$ corresponds to the case $N_{1}=A$, and the upper position $M_{2}$ corresponds to the case $L_{2}=C$.

To determine the positions of points $M_{1}$ and $M_{2}$ depending on the location of point $K$, we introduce a coordinate system as shown in Fig. 52.

Simple calculations show that if the abscissa of point $K$ is $x>0$, then $M_{1}=(-x, \sqrt{2 x})$, and $M_{2}=(-x, \sqrt{1-2 x}+1)$. Using the symmetry of the set of points $M$ with respect to the $O y$ axis, we find that this set forms a figure marked with hatching. There is no need to take integrals to calculate the area: in Fig. 51, the figures marked with the numbers 1 and 2 are equal.

![](https://cdn.mathpix.com/cropped/2024_05_21_7b8a6b2625543c3dae75g-140.jpg?height=369&width=264&top_left_y=381&top_left_x=151)

![](https://cdn.mathpix.com/cropped/2024_05_21_7b8a6b2625543c3dae75g-140.jpg?height=443&width=302&top_left_y=319&top_left_x=453)

Fig. 52

![](https://cdn.mathpix.com/cropped/2024_05_21_7b8a6b2625543c3dae75g-140.jpg?height=460&width=218&top_left_y=290&top_left_x=793)

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