The point $K$ lies in the interior of the unit circle. Four lines are drawn through $K$ such that each two adjacent lines form an angle of $45^{\circ}$, as shown in the picture. In such a way the circle is divided into 8 regions. Four of these regions are colored such that no two colored regions share more than one point. Find the maximal and minimal possible value for the total area of colored regions.

![](https://cdn.mathpix.com/cropped/2024_04_17_2f0af4b96a9a083248abg-2.jpg?height=414&width=417&top_left_y=381&top_left_x=842)

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We will show that the total area of colored regions is equal to $\frac{1}{2}$ the area of the circle no matter how the lines are drawn.

First we will inscribe our circle in a square whose sides are parallel to 2 of these lines. If we color the regions of the square in the same way as the regions of the circle are colored we can easily prove that the total area of colored regions is equal to the area of non-colored regions for the square. Hence it remains to prove the equality of the areas of colored and uncolored regions that are outside the cirle. Now we will look at the pictures below. For each picture it is obvious that it has the same area of colored parts as the previous picture (as some colored parts are just moved from one location to the other).
![](https://cdn.mathpix.com/cropped/2024_04_17_2f0af4b96a9a083248abg-2.jpg?height=342&width=756&top_left_y=1508&top_left_x=273)
![](https://cdn.mathpix.com/cropped/2024_04_17_2f0af4b96a9a083248abg-2.jpg?height=978&width=730&top_left_y=866&top_left_x=1095)