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Math Help - geometry question

  1. #1
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    Post geometry question

    hey everybody,
    im tring to solve this question for a long time without much success...
    i succeeded only with trigonemtry but this is not helping because i need to solve the question with geometry only!
    so here the question:


    The center of the squares are Q and P
    BO=OC
    I Need to prove OP=OQ

    Now, i found this question helpful:


    on each side of Parallelogram there is a square

    I need to prove that:
    If the centers of the squares are connectd
    a square forms in the middle
    if you can prove that the square in the middle has right angles
    then you can prove the first question (look at the green line its Symmetry line and also the hypotenuse)
    thank you for the help
    omer20
    Last edited by omer20; August 14th 2011 at 09:57 AM.
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  2. #2
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    Opalg's Avatar
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    Re: geometry question

    It looks as though you have made a very good start on this problem. You do not actually need to show that the "square in the middle" (which I assume is the thing with the blue diagonals) is a square. It will be sufficient to show that it is a rhombus, because the diagonal of a rhombus intersect at right angles. To do that, prove that the red triangles are congruent (two sides, included angle).

    In fact, it certainly seems to be true that the "square in the middle" really is a square, but I don't offhand see how to prove that.
    Attached Thumbnails Attached Thumbnails geometry question-rhombus.png  
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  3. #3
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    Re: geometry question

    hey
    thank you for the help
    i found a way to show that this is a square:

    if you can prove that the triangles are congruent (I couldnt find how to prove that one of the angles are even)
    you can see that there is a right angle in the middle and then by reducing angle and adding it again you can find that the square has a right angle and then the question is solved!
    but i cant find the angle so ill be glad if you write the angle you were talking about and how to prove that offcourse
    thank you!!!
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  4. #4
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    Re: geometry question

    Quote Originally Posted by omer20 View Post
    i found a way to show that this is a square:

    if you can prove that the triangles are congruent (I couldnt find how to prove that one of the angles are even) you can see that there is a right angle in the middle and then by reducing angle and adding it again you can find that the square has a right angle and then the question is solved!
    Very nice!

    Quote Originally Posted by omer20 View Post
    but i cant find the angle so ill be glad if you write the angle you were talking about and how to prove that offcourse
    Look at the obtuse angle in the two triangles. In each case it consists of three parts. Two of these parts are 45 angles (the angle between the side of a square and the diagonal). The part in the middle, in the left red triangle, is the acute angle of the parallelogram. In the right red triangle, you can see (from the fact that all the angles at that point add up to 360) that the part in the middle is the supplement of the obtuse angle of the parallelogram, and is therefore equal to the acute angle of the parallelogram.
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  5. #5
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    Re: geometry question

    thank you very much problem solved!!!
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  6. #6
    Senior Member abhishekkgp's Avatar
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    Re: geometry question

    Here's another solution which i think is simpler.

    K is mid-point of BP. L is mid point of CQ. D is mid-point of BC.

    this gives |CP|=2|DK|, \, |BQ|=2|DL|..
    Its easy to see that \Delta PAC \cong \Delta BAQ \Rightarrow |BQ|=|CP| \Rightarrow |DK|=|DL|.
    Attached Thumbnails Attached Thumbnails geometry question-img_0044.jpg  
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