let me try to explain.
the smith chart is a mapping of the REFLECTION COEFFICIENT PLANE.
to get reflection coefficient, physically, a signal has to start at a source, travel to a load, and then return to the source. you measure the return signal in relation to the original source signal.
So any result you get from a smith chart involves a wave travelling BOTH WAYS.
If the load is a half wave length away, the wave energy will pick up 180 degrees going from the source to the load, and another 180 degrees bouncing back from the load to the source.
the equations above are just saying that