Let $\Pi$ be an affine plane (An affine plane is a set of points and subsets called lines Satisfying (I1), (I2), (I3), (The axioms of Incidence) and the following stronger form of Playfair's axiom.
$P'$: For every line $l$, and every point A, there exists a unique line m containing A and parallel to $l$.
A pencil of parallel lines is the set of all the lines parallel to a given line (including that line itself). We call each pencil of paralleI lines an "ideal point," or a "point at infinity," and we say that an ideal point "lies on" each of the lines in the pencil. Now let $\Pi'$ be the enlarged set consisting of $\Pi$ together with all these new ideal points.
A line of $\Pi'$ will be the subset consisting of a line of $\Pi$ plus its unique ideal point, or a new line, called the "line at infinity," consisting of all the ideal points.
Question: I am asked to show that this new set $\Pi'$ with subsets of lines as just defined forms a projective plane.
This feels like when you draw something in $3D$ on a piece of paper one uses the concepts of line at infinity for things like tileings going off into the "horizon"
A projective plane is a set of points and subsets called lines that satisfy the following four axioms:
Pl. Any two distinct points lie on a unique line.
P2. Any two lines meet in at least one point.
P3. Every line contains at least three points.
P4. There exist three noncollinear points.
I believe we get P4 for free and P3 follows almost immediately from every line having 2 points and its ideal point laying on the same line which implys at least 3 points lay on every line .
Proving that each ideal point lies on a unique line im not sure and how to form P2 from playfairs axiom i am also not sure.