Let C_2k be the cycle on 2k vertices, and let ex(v, C_2k) denote the greatest number of edges in a simple graph on v vertices which contains no subgraph isomorphic to C_2k. In this paper we discuss a method which allows one to sometimes improve numerical constants in lower bounds for ex(v, C_2k). The method utilizes polarities in certain rank two geometries. It is applied to refute some conjectures about the values of ex(v, C_2k), and to construct some new examples of graphs having certain restrictions on the lengths of their cycles. In particular, we construct an infinite family {G_i} of C₆-free graphs with |E(G _i)| ∼ 1 2 |V(G _i)| ^{4 3} , i → ∞ , which improves the constant in the previous best lower bound on ex(v, C₆) from 2 3 ^{4 3} ≈ 0.462 to 1 2 .