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April 1st, 2015, 04:26 AM   #1
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Semicircle inscribed in trapezium

Hi so I'm stuck on another geometry problem:
ABCD is a trapezium with a semicircle, centre O, inscribed in it.*
If AO=OD, prove that (AO)(OD)=(AB)(CD).
I've tried Pythagoras' theorem on the radius, and extending AB and CD to meet but neither of them work. I'd appreciate any hints. Thanks.
*By centre of semicircle I mean centre of the full circle.
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April 1st, 2015, 05:06 AM   #2
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Can you provide a diagram?
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April 2nd, 2015, 02:09 AM   #3
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How can I provide a diagram?
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April 3rd, 2015, 12:02 AM   #4
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The diagram can be constructed as follows:
1. Construct a line segment AD and label its midpoint as O.
2. Choose 2 points, P,Q, along AD such that AP=DQ.
3. Construct a semicircle, centered at O, starting at P and ending at Q.
4. Let AM and DN be tangent to the semicircle at M and N respectively.
5. Construct point X where AM and DN extended meet.
6. Let BC be a tangent of the semicircle such that B is on AX and C on DX.
Then proceed to solve the problem.
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April 3rd, 2015, 02:30 AM   #5
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Are AD and BC parallel?
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