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October 12th, 2013, 07:14 AM  #1 
Newbie Joined: Oct 2013 Posts: 15 Thanks: 0  Proving a common point in the lines on the circle
We have a circle with a chord AB that is not a diameter. On the circle O choose the point P different from point A B. The points Q and R lie on the line PA and PB and QP = QB and RP=RA. Punkt M jest ?rodkiem odcinka QR (M is the midpoint of QR). Prove that all lines PM (depend on different positions point P on the circle O) have a common point.

October 13th, 2013, 02:51 AM  #2 
Newbie Joined: Oct 2013 Posts: 15 Thanks: 0  Re: Proving a common point in the lines on the circle
I know it's difficult, but maybe some ideas or hints ? 
October 13th, 2013, 10:08 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,746 Thanks: 2133 
As QP = QB, angle QBP = angle BPQ, and as RP = RA, angle BPQ = angle RAQ. Hence angle QBP = angle RAQ, and so ARBQ is a cyclic quadrilateral. Draw an accurate diagram and write down what else you can prove, even if it doesn't seem to help. Call the (apparent) common point S. Does PRSQ appear to be a parallelogram? (The above are just ideas, as I haven't yet found a proof that S exists.) 

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circle, common, lines, point, proving 
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