  We show that in the framework of CAT(0) spaces, any convex combination of two
mappings which are firmly nonexpansive -- or which satisfy the more general
property $(P_2)$ -- is asymptotically regular, conditional on its fixed point
set being nonempty, and, in addition, also $\Delta$-convergent to such a fixed
point. These results are established by the construction and study of a convex
combination metric on the Cartesian square of a CAT(0) space. We also derive a
uniform rate of asymptotic regularity in the sense of proof mining. All these
results are then interpreted in the special case of the mappings being
projections onto closed, convex sets.
