  We study the dynamical interaction in which the two single runaway stars AE
Aurigae and mu Columbae and the binary iota Orionis acquired their unusually
high space velocity. The two single runaways move in almost opposite directions
with a velocity greater than 100 km/s away from the Trapezium cluster. The star
iota Ori is an eccentric (e=0.8) binary moving with a velocity of about 10 km/s
at almost right angles with respect to the two single stars. The kinematic
properties of the system suggest that a strong dynamical encounter occurred in
the Trapezium cluster about 2.5 Myr ago. Curiously enough, the two binary
components have similar spectral type but very different masses, indicating
that their ages must be quite different. This observation leads to the
hypothesis that an exchange interaction occurred in which an older star was
swapped into the original iota Orionis binary. We test this hypothesis by a
combination of numerical and theoretical techniques, using N-body simulations
to constrain the dynamical encounter, binary evolution calculations to
constrain the high orbital eccentricity of iota Orionis and stellar evolution
calculations to constrain the age discrepancy of the two binary components. We
find that an encounter between two low eccentricity (0.4<e<0.6) binaries with
comparable binding energy, leading to an exchange and the ionization of the
wider binary, provides a reasonable solution to this problem.
