[Central symmetry helps solve the task]

Given angle $ABC$ and a point $D$ inside it. Construct a segment with endpoints on the sides of the given angle, the midpoint of which would be at point $D$.

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Let's construct points $A^{\prime}$ and $C^{\prime}$ as the intersections of the lines symmetric to lines $B C$ and $A B$ with respect to point $D$, with lines $A B$ and $B C$ (see figure). It is clear that point $D$ is the midpoint of the constructed segment $A^{\prime} C^{\prime}$, since points $A^{\prime}$ and $C^{\prime}$ are symmetric with respect to $D$.

![](https://cdn.mathpix.com/cropped/2024_05_06_ca19533aa595e185c5a8g-24.jpg?height=546&width=824&top_left_y=1482&top_left_x=617)

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