We consider the homogenized linear feasibility problem, to find an x on the unit sphere satisfying n linear inequalities a_i ^Tx \ge 0. To solve this problem we consider the centers of the inspheres of spherical simplices, whose facets are determined by a subset of the constraints. As a result we find a new combinatorial algorithm for the linear feasibility problem. If we allow rescaling, this algorithm becomes polynomial. We point out that the algorithm also solves the more general convex feasibility problem. Moreover, numerical experiments show that the algorithm could be of practical interest.