now we have a 'hexalateral' made up of six tangents to a conic. The lines joinng the opposite vertices meet at a point. This is regarded as the dual of Pascal's theorem, since lines and points are interchanged, and a point on a conic is replaced by a line tangent to it. This is a far from obvious duality, hence the century and a half between the two theorems we would now regard as one.

Pull the point D to deform the hexagon, and the point C to deform the conic. The point J moves but remains a triple junction. Moving the point D inside the hexagon makes all lines disappear, since the tangents now become imaginary - they are still there, somewhere where we cannot see them