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Math Help - Tricky geometry problem

  1. #1
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    Tricky geometry problem



    AB and DC are tangents to the small circle. I have proved that AD is parallel to BC.

    Prove that CD^2-ED^2 = ED x EB
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  2. #2
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    Re: Tricky geometry problem

    Hi,
    You really should be more explicit in exactly what problem you are posing. I had to guess at what you meant. The attachment shows what I think you want, but without details from you, I'm not absolutely certain. To follow the solution, you need to know about the power of a point with respect to a circle. If you don't know this, you should look it up.

    Tricky geometry problem-mhfgeometry62.png
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  3. #3
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    Re: Tricky geometry problem

    For a proof of that final equation, you don't need the point A nor the circle through A, B, C and D.

    One of the angle properties of a circle has that, since CD, (using your notation,) is a tangent to the circle, angle DCE is equal to the angle CBE.
    From that it follows that the triangles DBC and DCE are similar and by equating ratios of sides in those triangles the result follows.
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  4. #4
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    Re: Tricky geometry problem

    Thanks to BobP, I discovered a new (to me) proof of a fundamental property of the power of a point D with respect to a given circle, where D is outside the circle. The attachment shows this:

    Tricky geometry problem-mhfgeometry62a.png
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