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Math Help - Euclid's Elements - Book 1 Proposition 7

  1. #1
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    Euclid's Elements - Book 1 Proposition 7

    Geometry has always been a weakness of mine relative to algebra, so I decided to address this and picked up a copy of Euclid's Elements. I've managed to get as far as Proposition 7 of Book 1 before finding part of a proof that eludes me.

    I've attached the relevant image, and for those that don't have a copy of Elements to refer to, the key details are that sides AC and AD are equal, as are sides BC and BD, with the proof demonstrating that this construction is impossible.
    Euclid's Elements - Book 1 Proposition 7-prop7.gif
    I understand that angle ACD must equal angle ADC, and that angle ADC is greater than angle DCB. However, the proof then states that "angle CDB is much greater than the angle DCB" and I've not been able to satisfy myself that this must be true, only that it appears to be true for the triangles as drawn here. I assume there is some connection with a preceding proposition that I'm missing here, but I can't see what it is. If I just accept this step, the remainder of the proof makes sense to me, but obviously I want to understand the whole proof.

    Any help would be much appreciated.
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  2. #2
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    Re: Euclid's Elements - Book 1 Proposition 7

    Since BC = BD, then angle BCD = angle BDC.
    But, angle BCD is less than angle ACD (from the diagram !)
    while angle BDC is greater than angle ADC (again from the diagram).
    However, since angle ACD = angle ADC we have a contradiction, in which case the given diagram is impossible.
    Thanks from britmath
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