Given are the coordinates (x;y) of five points on a non-degenerate conic section. I know how to determine the type (circle, ellipse, parabola, hyperbola) and parameters (location of center, foci, rotation angle, a, b, c) using the general equation $A\cdot x^2+B\cdot x\cdot y+C\cdot y^2+D\cdot x+E\cdot y+1=0$ and matrix calculation.
Now I wonder if there is another, geometric method when the axes and location of the center and foci are unknown.
My question: given the location of five points in a diagram, how can I determine the type of the conic section using a ruler, compass (and if necessary, minimal calculations)?
