Say I have a line segment $AB$. I have a compass that can only create a circle with some random radius $r$ that is less than the length of $AB$. (I also have a straight-edge to create lines of arbitrary length.)
In this case, is there a general method for constructing an equilateral triangle with $AB$ as one side, using only Euclid's 5 postulates and common mathematical notions(ex: if $a=b$ and $b=c$ then $a=c$)?
Start by drawing a point $B'$ on the $AB$ segment, at a distance $r<AB$ from $A$, towards $B$. You know how to construct $C'$ say above $AB$, such that $AB'C'$ is an equilateral triangle with side $r$. Note that $\angle B'AC'=\angle BAC'=60^\circ$. Now repeat the same procedure at point $B$: find $A''$ towards $A$ on $AB$, at distance $r$ from $B$, then construct $C''$ so that the triangle $A''BC''$ is equilateral, with $C''$ on the same side of $AB$ as $C'$. Once again, notice that $\angle A''BC''=\angle ABC''=60^\circ$. Now all you need to do is extend $AC'$ and $BC''$ until they meet at a point $C$. Since two of the angles in the $ABC$ triangle are $60^\circ$, you have an equilateral triangle, with side $AB$.