Problem 3. Given segments $a, b, c$. Construct a segment of length $\frac{a b}{c \sqrt{5}}$ using a compass and straightedge.

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# Solution.

1) The segment of length $c \sqrt{5}$ is constructed as the hypotenuse of a right triangle with legs $c$ and $2c$.
2) The segment of length $x=\frac{a b}{c \sqrt{5}}$ is constructed according to Thales' theorem as a proportional segment, cut off on the sides of an angle

![](https://cdn.mathpix.com/cropped/2024_05_06_67444cb6f337a86f9dddg-1.jpg?height=242&width=323&top_left_y=1815&top_left_x=1620)
by parallel lines from the ratio $\frac{X}{b}=\frac{a}{c \sqrt{5}}$.