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How to Find Collaborators to Assist with your Project both within and outside of your field?

    There are times when you are working on a project and discover that, in order to complete the project, you need to bring a collaborator on board who is an expert in an area adjacent to yours.  There are a few ways to handle this.  
    Sometimes you just need a theorem from another field and can be informed of an existing theorem by a colleague and just provide an acknowledgement.  Other times, you may need a new theorem proven in another field.  In this case, you may find someone who is a postdoc somewhere and suggest the problem to him or her.  If the theorem is interesting enough, they get a paper which will then be cited by your paper.  If the theorem doesn't stand alone, they can write an appendix for your paper.   Notice that these three methods allow you to believe a theorem from an adjacent field which you may or may not be able to verify and which you contribute nothing towards solving.   In my experience the appendix approach is fine if the person is not going up for tenure, but if they are, it can hurt them as they are not getting a paper.   
    Sometimes you need to really learn another field in order to apply the field in combination with your own to complete a paper.  This requires finding a coauthor in the adjacent field who knows a little about your own field and working together intensively.   Again, a postdoc at a nearby university is often a good candidate because they are eager and willing to learn.  You need to make sure the person is an expert in their field and that he or she can communicate with patience.   Inviting the person to speak and asking lots of questions will give you some sense of how well the person can answer questions.  Only after you are certain the person can communicate well with you, should you consider mentioning the joint project.   
See also Krishnan Shankar's response to III.C. which includes a discussion of how to handle a frustrated coauthor.