Prove line in same plane as triangle that intersects interior, intersects a side

Please let me know if the following proof is correct or not. If not a hint would be appreciated.

Given and line *l* lying in the same plane as . Prove that if *l *intersects the INT( ) then it intersects at least one of the sides.

Proof: Let D and E be points on *l* and D,E Since D,E they are on the same side of as B. There are seven possibilities:

1) forms a line such that A,B,C are all on the same side of

2) A is on Then we are done by the crossbar theorem.

3) B is on Then we are done by the crossbar theorem.

4) C is on Then we are done by the crossbar theorem.

5) A is on the opposite side of as B and C. Then by the plane separation theorem intersects and intersects

6) B is on the opposite side of as A and C. Then by the plane separation theorem intersects and intersects

7) C is on the opposite side of as A and B. Then by the plane separation theorem intersects and intersects

For 1), we have already determined that since D,E they are on the same side of as B. If A, B, and C were all on the same side of this would be a contradiction. So, 1) cannot be true. Therefore one of the other conditions must be true. All of the other conditions have *l* intersecting a side of the triangle so the theorem is proved.