I have come up with a possible proof. I used proof by contradiction and Incidence Axiom 2 from my textbook, which states that "For every line there exists at least two distinct points on it". Please check the proof for accuracy.
Proof: Let c and b be two lines on an affine plane which intersect at a point P. Let a be a line that is parallel to line b. We want to show that line a intersects line c. Suppose to the contrary that line a is parallel to line c. This means that line a does not contain point P. We have that both lines a and b pass through point P and both lines a and b are parallel to line c. However, this contradicts Incidence Axiom 2 since line c does not contain 2 distinct points. Therefore, line a intersects line c, which concludes the proof.
Your argument states that line a does not contain point P, and then that line a passes through point P, so this cannot be a correct proof. The rough outline of the proof is as follows (I'll let you fill in the reasons each assertion is correct):