Let A and B be two fixed points on a given circle and M and N the ends of any diameter. Find the locus of the point of intersection of lines AM and BN as MN rotates about the center of the circle.
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Prove or disprove: The locus the point of intersection of AM and BM is a circle when AB is a chord that is not a diameter.
Question: Where is the center of the circle and what is its radius?
Extension: What happens as A and B are placed at (near) the ends of a diameter?