The locus of the point at which two given circles possess the same circle poweris a straight line perpendicular to the line joining the midpoints of the circle and is known as the chordal (or, more commonly, the radical line) of the two circles.

Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 153, 1965.

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Weisstein, Eric W. "Chordal Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ChordalTheorem.html