Suppose that in the affine plane R^2 four lines are given, with the property that no two are parallel and no three are concurrent. Show that there exists a unique parabola tangent to each of the four lines.
My attempts: I want to extend the affine plane R^2 to a projective plane PR^2 and consider the dual space of PR^2. But 'parallel' and 'being a parabola' are not projective properties, so things become difficult.