The focus of this chapter is twofold. The first is a semiotic description of the nature of diagrams. The second is a description of the type of reasoning that the transformation of diagrams facilitates in the construction of mathematical meanings. I am guided by the Peircean definition of diagrams as icons of possible relations and his conceptualization of diagrammatic reasoning. When a diagram is actively and intentionally observed, perceptually and intellectually, a manifold of structural relations among its parts emerges. Such relations among the parts of the diagram can potentially unveil the deep structural relations among the parts of the Object that the icon plays to represent. An Interpreter, who systematically observes and experiments with diagrams, mathematical or not, also generates evolving chains of interpretants by means of abductive, inductive and deductive thinking. Using Stjernfelt's model of diagrammatic reasoning, which is rooted in Peircean semiotics, I illustrate an emergent reasoning process to prove two geometric propositions that were posed by means of diagrams.