Note on $120$ degree axial coordinates

Recently the author introduced [1] a symmetrical system of distance coordinates for studying the planar motion of a three-body system of point masses in its internal degrees of freedom, and applied them to the small vibration problem. Subsequently he applied them [2] to the gravitational three-body problem. The purpose of this note is to give a geometrical identification of the auxiliary angles ${\theta _{1,}}{\theta _{2,}}$, and ${\theta _3}$, related to the system, and to prove the identity which they satisfy.