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 April 1st, 2015, 04:26 AM #1 Newbie   Joined: Dec 2014 From: Singapore Posts: 23 Thanks: 0 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.
 April 1st, 2015, 05:06 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,978 Thanks: 2229 Can you provide a diagram?
 April 2nd, 2015, 02:09 AM #3 Newbie   Joined: Dec 2014 From: Singapore Posts: 23 Thanks: 0 How can I provide a diagram?
 April 3rd, 2015, 12:02 AM #4 Newbie   Joined: Dec 2014 From: Singapore Posts: 23 Thanks: 0 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.
 April 3rd, 2015, 02:30 AM #5 Global Moderator   Joined: Dec 2006 Posts: 20,978 Thanks: 2229 Are AD and BC parallel?

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### A trapezium with a semi circle

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