Find locus of midpoint in circle intersections

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The discussion centers on finding the locus of the midpoint of a line segment PAQ formed by points P and Q on two intersecting circles at points A and B. A participant suggests that the midpoint M of PAQ is related to the centers of the circles and that M coincides with point A when the line OM is perpendicular to PAQ. They reference basic geometric properties learned in middle school but express difficulty in progressing further with the solution. The conversation invites additional methods or modifications to tackle the problem effectively. Overall, the focus remains on geometric constructions and the properties of circle intersections.
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Two circles intersect at points A and B. PAQ is a straight line such that points P and Q lie on the two circles. Find the locus of the midpoint of PAQ
 
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Please, tell us what you have done so far in solving the problem. :)
 
I have tride with saying that if c and d are the centres of the circle and o is its midpoint of CD then by structure as well as by sense we can feel that M is midpoint of PAQ where OA = OM. thus midpoint lies at A when OM is perpendicular to PAQ. these are basic construction properties which i have noted in middle school. but cannot proceed furhter. any other way or modification to this
 

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