A three-way concept lattice offers much richer information than a classical concept lattice, since it can extract positive information and negative information from a formal context simultaneously. While for the same formal context, the structure of its three-way concept lattice is usually more complex than that of the corresponding classical concept lattice. The observation reflects the fact that we need to pay much more cost for the three-way concept lattice construction to seek additional information. Therefore how to obtain abundant information from a three-way concept lattice with the same structure as a classical concept lattice deserves to be studied. Driven by this issue, we introduce two kinds of specific formal contexts, namely, attribute-induce dual formal context (AID-formal context) and attribute-induce dual intersection formal context (AIDI-formal context). On this basis, we investigate the isomorphic relation between classical concept lattices and three-way concept lattices of the proposed formal contexts. Specifically, the sufficient conditions and necessary conditions for the isomorphic relation between classical concept lattices and three-way concept lattices are studied, respectively. At last, the algorithms of judging AID-formal context and AIDI-formal context are given, and the number of AID-formal context and AIDI-formal context in randomly generated formal contexts is compared. It is of great significance to explain the relationship between them and to put forward three-way concept lattices.