(1) If the circle has center at C (1,0) then A and B will be equidistant. If the circle say has center at (0,1) and passes through origin and A is (0,0) and B is (0,2) then A and B are NOT equidistant from C. Hence Insufficient.

(2) The question says length bisector, not perpendicular bisector. So, there can be many combinations of points A and B such that the mid point of AB has a line passing through C. Cannot say if the points are equidistant. Hence Insufficient.

Combined: In the case of the first circle with center C (1,0), the length bisector of AB will always pass through C. But, in case of the second circle with center (0,1) and A being (0,0) and B being (0,2), there will still be a line which passes through midpoint of AB and C, but A and B are not equidistant to C. Hence Insufficient.

Hence IMO, Option E

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