\[ct'=\frac{ct_0}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\frac{ct}{2}\sqrt{1-\frac{v^{2}}{c^{2}}}=ct_1\sqrt{\frac{1-\frac{v}{c}}{1+\frac{v}{c}}}=ct_2\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}=c\sqrt{t_1\cdot t_2}=c\sqrt{\frac{t\cdot t_0}{2}}=\frac{l_0}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\]