If we are limited by what we can construct with compass and straight edge, then what becomes possible by expanding our toolkit to include all conic sections? The tools would be based on already constructed points, lines and circles.
I'm aware that the hyperbola allows for angle trisection and the construction of a heptagon; however, I have not read anything about a conic section being used to "square the circle".
Does an ellipse or parabola contribute anything?
To be clear, my interest is not about methods for "squaring the circle" and such; it's about what the conic sections do and do not add to the realm of possibilities regarding geometric construction.