If the length of the string is fixed (i.e. if the string is inextensible), then we have:
($l$ is the length of the string)
$$l=(s_1-s_3)+(s_1-s_5)+\pi R=2s_1-s_3-s_5+\pi R$$ $$l=(s_2-s_4)+(s_2-s_6)+\pi R=2s_2-s_4-s_6+\pi R$$ $$z=s_2-s_1\quad x=s_4-s_3\quad y=s_6-s_5$$ $$\Longrightarrow\; 2s_2-2s_1=(s_4-s_3)+(s_6-s_5)$$ $$\Longrightarrow\; 2z=x+y$$