  The classical convex feasibility problem in a finite dimensional Euclidean
space is studied in the present paper. We are interested in two cases. First,
we assume to know how to compute an exact project onto one of the sets involved
and the other set is compact such that the conditional gradient (CondG) method
can be used for computing efficiently an inexact projection on it. Second, we
assume that both sets involved are compact such that the CondG method can be
used for computing efficiently inexact projections on them. We combine
alternating projection method with CondG method to design a new method, which
can be seen as an inexact feasible version of alternate projection method. The
proposed method generates two different sequences belonging to each involved
set, which converge to a point in the intersection of them whenever it is not
empty. If the intersection is empty, then the sequences converge to points in
the respective sets whose distance is equal to the distance between the sets in
consideration.
