I'll do this for the parabola since I'm more familiar with that. But the proof applies to any parabola.

Take the three points to be The area of the triangle PQR is half the absolute value of Using elementary row operations and expanding down the right-hand column, this comes out to be

The tangent at R is . It meets the tangent at S at the point . Similarly, and The area of the triangle P'Q'R' is half the absolute value of

Thus area(PQR) = 2area(P'Q'R').