4. Let $k$ be a circle with diameter $A B$, and let $C$ be an arbitrary point on the segment $A B$. Find two points $X$ and $Y$ on the circle $k$ that are symmetric with respect to $A B$ and such that the lines $A X$ and $C Y$ are perpendicular to each other.
Solution. Let $X$ be the drawn point and let

$$
A X \cap C Y=M, A B \cap X Y=N
$$

Then

$$
\measuredangle A M Y=90^{\circ} \text { and } \measuredangle A X B=90^{\circ},
$$

so $B X \| C Y$. In addition, $\overline{N X}=\overline{N Y}$ and $B C \perp X Y$, so $N$ is the midpoint of $B C$.

## II year

![](https://cdn.mathpix.com/cropped/2024_06_05_47c254b2217fdab00c02g-2.jpg?height=420&width=468&top_left_y=809&top_left_x=1011)