On projectively invariant points of an oval with a distinguished exterior line

We consider projectively invariant points of an oval with a distinguished exterior line. For this, we introduce a projectively invariant transformation of the line parametrized by the oval. Projectively invariant points are defined as fixed points of this transformation applied twice. We prove that there are at least four such points. For the proof we reduce the problem to an affine problem and construct an extremal area parallelogram circumscribed around the oval.