Conic5PointsTwo points determine a line, three a circle, and it takes five points to determine a conic. In this sketch, the 5 points are A, B, C, D, E. Drag these points, and you'll see their conic.
This construction is based on a theorem that Pascal discovered while a teenager. The conic is sketched as the locus of the point F.
As you drag the 5 points, why do you see lots more ellipses and hyperbolas (with asymptotes) than parabolas?
If 4 of the points are vertices of a rectangle, where must the 5th point be in order for the conic to be an ellipse?
For more, see GEOMETRY IN ACTION, Chapter 7.