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Doubling a cube and trisecting any angle
No, it is not. In 1837, the French mathematician, Pierre Laurent Wantzel, proved that it was impossible to do so using only compass and straightedge.
No, it is not and in 1837 Pierre Wantzel proved this to be the case.
No. This is known to be impossible. For more information, including a proof, check the Wikipedia article on "doubling the cube".
Yes, it is possible.