Trisecting agents, issues, and agent pairs are essential topics of three-way conflict analysis. They have been commonly studied based on either a rating or an auxiliary function. A rating function defines the positive, negative, or neutral ratings of agents on issues. An auxiliary function defines the alliance, conflict, and neutrality relations between agents. These functions measure two opposite aspects in a single function, leading to challenges in interpreting their aggregations over a group of issues or agents. For example, when studying agent relations regarding a set of issues, a standard aggregation takes the average of an auxiliary function concerning single issues. Therefore, a pair of alliance +1 and conflict -1 relations will produce the same result as a pair of neutrality 0 relations, although the attitudes represented by the two pairs are very different. To clarify semantics, we separate the two opposite aspects in an auxiliary function into a pair of alliance and conflict functions. Accordingly, we trisect the agents, issues, and agent pairs and investigate their applications in solving a few crucial questions in conflict analysis. Particularly, we explore the concepts of alliance sets and strategies. A real-world application is given to illustrate the proposed models.