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Math Help - Check my proof about perpendicular lines which are parallel

  1. #1
    Senior Member OReilly's Avatar
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    Check my proof about perpendicular lines which are parallel

    I have to prove this theorem:

    If two different lines a and b are perpendicular to plane \alpha then a and b are parallel lines.

    My proof:
    Lines a and b are perpendicular on plane \alpha in points A and B.

    Through points A and B there is line c which is perpendicular to lines a and b.
    Lines b and c form plane \beta perpendicular to line d (in point B).
    Through point B there is line e which intersects line a and is normal to line d.
    Lines e and b form plane \gamma which is normal to line d, but then there would exist two planes that contains point B and are perpendicular to line d which is possible only if \beta=\gamma.
    Then plane \beta would contain also lines c and e and also line a (because lines c and e intersects line a) which means that lines a and b are both in plane \beta.

    Lines a and b don't intersect because than it would exist two lines perpendicular to line c in one point (intersection point) so they are parallel.


    Is my proof ok?
    Attached Thumbnails Attached Thumbnails Check my proof about perpendicular lines which are parallel-image.gif  
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by OReilly View Post
    I have to prove this theorem:

    If two different lines a and b are perpendicular to plane \alpha then a and b are parallel lines.

    My proof:
    Lines a and b are perpendicular on plane \alpha in points A and B.

    Through points A and B there is line c which is perpendicular to lines a and b.
    Lines b and c form plane \beta perpendicular to line d (in point B).
    Through point B there is line e which intersects line a and is normal to line d.
    Lines e and b form plane \gamma which is normal to line d, but then there would exist two planes that contains point B and are perpendicular to line d which is possible only if \beta=\gamma.
    Then plane \beta would contain also lines c and e and also line a (because lines c and e intersects line a) which means that lines a and b are both in plane \beta.

    Lines a and b don't intersect because than it would exist two lines perpendicular to line c in one point (intersection point) so they are parallel.


    Is my proof ok?
    Looks okay to me, but if I may just say that reading the title of this post made my morning!

    -Dan
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