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

  1. #1
    Member oldguynewstudent's Avatar
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    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 \triangle ABC and line l lying in the same plane as \triangle ABC. Prove that if l intersects the INT( \triangle ABC) then it intersects at least one of the sides.
    Proof: Let D and E be points on l and D,E \in INT(\triangle ABC). Since D,E \in INT(\triangle ABC) they are on the same side of \overleftrightarrow{AC} as B. There are seven possibilities:
    1) \overleftrightarrow{DE} forms a line such that A,B,C are all on the same side of \overleftrightarrow{DE}.
    2) A is on \overleftrightarrow{DE}. Then we are done by the crossbar theorem.
    3) B is on \overleftrightarrow{DE}. Then we are done by the crossbar theorem.
    4) C is on \overleftrightarrow{DE}. Then we are done by the crossbar theorem.
    5) A is on the opposite side of \overleftrightarrow{DE} as B and C. Then by the plane separation theorem \overleftrightarrow{DE} intersects \overline{AB} and intersects \overline{AC}.
    6) B is on the opposite side of \overleftrightarrow{DE} as A and C. Then by the plane separation theorem \overleftrightarrow{DE} intersects \overline{AB} and intersects \overline{BC}.
    7) C is on the opposite side of \overleftrightarrow{DE} as A and B. Then by the plane separation theorem \overleftrightarrow{DE} intersects \overline{BC} and intersects \overline{AC}.
    For 1), we have already determined that since D,E \in INT(\triangle ABC) they are on the same side of \overleftrightarrow{AC} as B. If A, B, and C were all on the same side of \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 have l intersecting a side of the triangle so the theorem is proved.
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  2. #2
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    I think that your proof works. Nonetheless, here are some comments.
    I am not sure that you need two points on the line interior to the triangle.
    If any one of A, B or C is on the line you are done.
    If assuming that A, B, & C are on the same side of the line leads to a contradiction then you are also done.
    Last edited by Plato; October 6th 2010 at 06:12 AM.
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