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

Given $\displaystyle \triangle ABC $ and linellying in the same plane as $\displaystyle \triangle ABC$. Prove that iflintersects the INT($\displaystyle \triangle ABC$) then it intersects at least one of the sides.

Proof: Let D and E be points onland D,E $\displaystyle \in INT(\triangle ABC). $ Since D,E $\displaystyle \in INT(\triangle ABC)$ they are on the same side of $\displaystyle \overleftrightarrow{AC}$ as B. There are seven possibilities:

1) $\displaystyle \overleftrightarrow{DE}$ forms a line such that A,B,C are all on the same side of $\displaystyle \overleftrightarrow{DE}.$

2) A is on $\displaystyle \overleftrightarrow{DE}. $ Then we are done by the crossbar theorem.

3) B is on $\displaystyle \overleftrightarrow{DE}. $ Then we are done by the crossbar theorem.

4) C is on $\displaystyle \overleftrightarrow{DE}. $ Then we are done by the crossbar theorem.

5) A is on the opposite side of $\displaystyle \overleftrightarrow{DE}$ as B and C. Then by the plane separation theorem $\displaystyle \overleftrightarrow{DE}$ intersects $\displaystyle \overline{AB}$ and intersects $\displaystyle \overline{AC}.$

6) B is on the opposite side of $\displaystyle \overleftrightarrow{DE}$ as A and C. Then by the plane separation theorem $\displaystyle \overleftrightarrow{DE}$ intersects $\displaystyle \overline{AB}$ and intersects $\displaystyle \overline{BC}.$

7) C is on the opposite side of $\displaystyle \overleftrightarrow{DE}$ as A and B. Then by the plane separation theorem $\displaystyle \overleftrightarrow{DE}$ intersects $\displaystyle \overline{BC}$ and intersects $\displaystyle \overline{AC}.$

For 1), we have already determined that since D,E $\displaystyle \in INT(\triangle ABC)$ they are on the same side of $\displaystyle \overleftrightarrow{AC}$ as B. If A, B, and C were all on the same side of $\displaystyle \overleftrightarrow{DE}, $ this would be a contradiction. So, 1) cannot be true. Therefore one of the other conditions must be true. All of the other conditions havelintersecting a side of the triangle so the theorem is proved.