"Foundations of Projective Geometry" by Hartshorne says the following:
The completion of the affine plane of four points is a projective plane with 7 points.
The affine plane of $4$ points is essentially a paralellogram $ABCD$. The completion will contain $A,B,C,D,[AB],[AD],[AC],[BD]$. Here $[AC]$ is the point of intersection of all lines parallel to $AC$ with the line at infinity (in other words it is an ideal point).
Hence I am getting $8$ points instead of $7$. Where am I going wrong?