Proof: By Euclid
(related to Proposition: 1.03: Cutting a Segment at a Given Size)
 Let the line $AD$, equal to the straight line $C$, have been placed at point $A$ [Prop. 1.2].
 And let the circle $DEF$ have been drawn with center $A$ and radius $AD$ [Post. 3] .
 And since point $A$ is the center of circle $DEF$, $AE$ is equal to $AD$ [Def. 1.15] .
 But, $C$ is also equal to $AD$.
 Thus, $AE$ and $C$ are each equal to $AD$.
 So $AE$ is also equal to $C$ [C.N. 1] .
 Thus, for two given unequal straight lines, $AB$ and $C$, the (straight line) $AE$, equal to the lesser $C$, has been cut off from the greater $AB$.
 (Which is) the very thing it was required to do.
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"