  We continue the study of infinite geodesics in planar first-passage
percolation, pioneered by Newman in the mid 1990s. Building on more recent work
of Hoffman, and Damron and Hanson, we develop an ergodic theory for infinite
geodesics via the study of what we shall call `random coalescing geodesics'.
Random coalescing geodesics have a range of nice asymptotic properties, such as
asymptotic directions and linear Busemann functions. We show that random
coalescing geodesics are (in some sense) dense in the space of geodesics. This
allows us to extrapolate properties from random coalescing geodesics to obtain
statements on all infinite geodesics. As an application of this theory we solve
the `midpoint problem' of Benjamini, Kalai and Schramm and address a question
of Furstenberg on the existence of bigeodesics.
