The role of string in constructive geometry

I was wondering whether, if I add string and thumbtacks to my geometry kit, I am able to do any new constructions. The idea being, with string, I can draw ellipses for instance and the intersection of these with lines, circles and other ellipses might produce points which aren't constructable from compass and straightedge alone. But then again, maybe not? This is why I'm asking. Thanks!

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We are assuming that for Euclidean constructions we are initially given two points $A$ and $B$. We identify $A$ with the origin, $B$ with $(1,0)$, and draw axes as usual. Then the two ellipses with equations $$2x^2-2x+y^2+2\sqrt{2}y+1=0\qquad\text{and}\qquad 3x^2-2x+y^2 +(2\sqrt{2}+1)y+\sqrt{2}=0$$ are string constructible. Subtract. We obtain $y=-(x^2+\sqrt{2}-1)$. Substitute for $y$ in the first equation, and simplify. We get $x^4-2x=0$. So one of the intersection points has $x$-coordinate $\sqrt[3]{2}$, and we have duplicated the cube.
Remark: I think I can make $4x^3-3x-\cos\theta=0$ pop out of the intersection of two string constructible ellipses, where $\cos\theta$ is a constructible number, through the kind of fooling around that produced $x^3-2=0$.