  It it shown that each lobe of the figure-eight orbit is star-shaped, which
implies the polar angle is monotone in each lobe. In general, it is not clear
when a minimizer is star-shaped. In this paper, we study minimizers connecting
two fixed-ends (i.e. the Bolza problem) in the planar three-body problem with
two equal masses. We show that if the Jacobi coordinates of the two fixed-ends
are in adjacent closed quadrants, then the corresponding minimizer must stay in
two adjacent closed quadrants. If we further assume the two Jacobi coordinates
are orthogonal on one of the fixed-ends, then the polar angles of the Jacobi
coordinates in the minimizer have at most one critical point. If the two Jacobi
coordinates are orthogonal on both ends, then the two polar angles must be
monotone. These geometric properties can be applied to show the existence of
two sets of periodic orbits.
