Distance between two circles

Question

See image. In situation like this. Is AE and EC always equal. if no then in which cases are the two lines equal.

Answer 1

No they are not equal in general. They are only equal if the circles have the same radius. If the circles do have the same radius, you can prove that the triangles $\triangle AED$ and $\triangle CED$ are congruent.

If they are congruent then $AE= EC$. The congruence requires $AB = CB$, so the two radiuses have to be the same.

Answer 2

Since $AB=AD$ and $CB=CD$, we obtain $BE=ED$ and if you want that also $AE=EC$ then $ABCD$ would be parallelogram, which says $AB=CD.$