Geometry · Intersecting circles · Thales theorem · Perpendicular bisector · Trapezoid midline · Fixed point
Two circles \(R_1\) and \(R_2\) meet at points \(A\) and \(B\). A line through \(A\) is allowed to vary; it meets \(R_1\) again at \(P\) and \(R_2\) again at \(Q\). Prove that all the resulting perpendicular bisectors of the segments \(PQ\) pass through one and the same point.
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Serbian Republic Competition (Republicko takmicenje) 1997, high school grade I, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source