Projective invariants are not only widely used in many practical applications, but also have close associations with classical algebraic geometry. In this work, we discover a novel projective invariant named as characteristic number, from which we obtain an intrinsic property of an algebraic hypersurface involving the intersections of this hypersurface and some lines that constitute a closed loop. As applications of this property, we show two high-dimensional generalizations of Pascal's theorem, one establishing the connection of hypersurfaces of distinct degrees, and the other concerned with the intersections of a hypersurface and a simplex.