Proof: By Euclid

Let a diameter $BC$ of circle $ABC$ have been drawn.¹ Therefore, if $BC$ is equal to $D$ then that (which) was prescribed has taken place.
For the (straight line) $BC$, equal to the straight line $D$, has been inserted into the circle $ABC$.
And if $BC$ is greater than $D$ then let $CE$ be made equal to $D$ [Prop. 1.3], and let the circle $EAF$ have been drawn with center $C$ and radius $CE$.
And let $CA$ have been joined.
Therefore, since the point $C$ is the center of circle $EAF$, $CA$ is equal to $CE$.
But, $CE$ is equal to $D$.
Thus, $D$ is also equal to $CA$.
Thus, $CA$, equal to the given straight line $D$, has been inserted into the given circle $ABC$.
(Which is) the very thing it was required to do.²

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Presumably, by finding the center of the circle [Prop. 3.1], and then drawing a line through it (translator's note). ↩
It has been shown that it is possible to construct a chord of a circle with a length less or equal to the diameter. ↩