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: 20,633 Thanks: 2080 
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: 20,633 Thanks: 2080 
Are AD and BC parallel?


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