A non-endpoint of the Cantor ternary set is any Cantor point which is not an endpoint of one of the remaining closed intervals obtained in the usual construction process of the Cantor ternary set in the unit interval. It is shown that the set of points in the unit interval which are not midway between two distinct Cantor ternary points is precisely the set of Cantor nonendpoints. It is also shown that the generalized Cantor set C λ , for 1 / 3 < λ < 1 , has void intersection with its set of midpoints obtained from distinct members of C λ .