Binary cyclic codes are worth studying due to their applications and theoretical importance. It is an important problem to construct an infinite family of cyclic codes with large minimum distancedand dual distanced⊥. In recent years, much research has been devoted to improving the lower bound ond, some of which have exceeded the square-root bound. The constructions presented recently seem to indicate that when the minimum distance increases, the minimum distance of its dual code decreases. In this paper, we focus on the new constructions of binary cyclic codes with lengthn=2m− 1, dimension nearn/2 and both relatively large minimum distance and dual distance. Whenmis even, we construct a family of binary cyclic codes with parameters [2m− 1, 2m−1± 1, d], whered≥ 2m/2− 1 andd⊥≥ 2m/2. Both the minimum distance and the dual distance are significantly better than the previous results. Whenmis the product of two distinct primes, we construct some cyclic codes with dimensionsk= (n+1)/2 andd>n/log2n, where the lower bound on the minimum distance is much larger than the square-root bound. Whenmis odd, we present two families of binary [2m−1, 2m−1,d] cyclic codes withd≥ 2(m+1)/2−1,d⊥≥2(m+1)/2andd≥ 2(m+3)/2− 15,d⊥≥ 2(m−1)/2respectively, which leads thatd·d⊥can reach 2nasymptotically. To the best of our knowledge, except for the punctured binary Reed-Muller codes, there is no other construction of binary cyclic codes that reaches this bound.