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I know that you can construct a square root of a given segment through a certain construction, and I know that construction. What is the proof that this construction works to construct a square root?

CsunWG _
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1 Answers1

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The usual construction is to make line segments of length $1$ and $x$ on the diameter a semicircle, and extend a vertical line segment up to meet the semicircle. Say the new line segment has length $s$. Then using similar triangles you can show that $$1:s :: s:x,$$ i.e., $$\frac1s = \frac sx,$$ so that $s^2 = x$.

Théophile
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