Two circles of unequal radii are drawn apart from each other. A point X is taken outside both of them, and the tangent drawn from X to the first circle turns out to have exactly the same length as the tangent drawn from X to the second. Considering every point X outside both circles for which this happens, what shape do those points form?

The squared tangent length from X is the power of the point: XO₁² − r₁² for the first circle and XO₂² − r₂² for the second. Equal tangents therefore mean XO₁² − XO₂² = r₁² − r₂², a fixed number. The set of points whose squared distances to two fixed points differ by a constant is a straight line at right angles to the segment joining them, so X runs along such a line (the radical axis). The tempting answer is the perpendicular bisector, but that is the special case r₁ = r₂; with unequal radii the line is pushed towards the smaller circle. A parallel line would give a varying difference, and no circle has this property.