The Maia fixed point theorem is one of the interesting generalizations of the well-known Banach contraction principle. In this manuscript, we introduce two notions called mixed $UC-$property and mixed $P-$property of a pair $(A,B)$ of nonempty subsets of a set $X$ endowed with two metrics. First, we consider a cyclic mapping $T: A\cup B \rightarrow A\cup B$, where A and B are nonempty subset of a set $X$ endowed with two metrics, and using the mixed UC Property we obtain sufficient conditions for the existence of best proximity points of T. When A=B, then our result reduces to the Maia's fixed point theorem. Secondly, we consider a nonself mapping $T: A \rightarrow B$, where A and B are nonempty subset of a set $X$ endowed with two metrics, and using mixed P-property we obtained sufficient conditions for the existence of best proximity points of $T$. Thus, we present two best proximity point theorems which generalize the Maia fixed point theorem.