| [ Properties and characteristics of an isosceles triangle. Trigonometric relationships in a right triangle |
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In an isosceles triangle, the angle at the vertex is $\alpha$, and the area is $S$. Find the base.
![](https://cdn.mathpix.com/cropped/2024_05_06_4fce0705b02fa9e106d4g-24.jpg?height=66&width=2026&top_left_y=1&top_left_x=-1)
$a^{2} /{ }_{4} \operatorname{ctg} \alpha / 2$. From here $a=2 \sqrt{S \operatorname{tg} \frac{\alpha}{2}}$.

## Answer

## $2 \sqrt{S \operatorname{tg} \frac{\alpha}{2}}$.