The formation of a liquid bridge between a cone and a plane is related to dip-pen nanolithography. The meniscus shape and rupture process of a liquid meniscus between a cone and a plane are investigated by Surface Evolver, many-body dissipative particle dynamics, and macroscopic experiments. Dependent on the cone geometry, cone-plane separation, and wetting properties of cone and plane, three types of menisci can be observed before rupture and two types of wetting competition outcomes are seen after breakup. It is interesting to find that after rupture, the bulk of the liquid bridge volume is not necessarily retained by the cone which is more wettable. In fact, a sharp hydrophilic cone often loses wetting competition to a hydrophobic plane. To explain our findings, the "apparent" contact angle of the cone is introduced and the behavior of drop-on-cone/plane system is analogous to that of a liquid bridge between two parallel planes based on this concept.