Let the angle between two nonzero vectors $\overrightarrow{A}\text{and}\overrightarrow{B}$ be $120°$ and its resultant be $\overrightarrow{\text{C}}$

$\left|\overrightarrow{A}+\overrightarrow{B}\right|=\left|\overrightarrow{C}\right|$

$\sqrt{{\text{A}}^{\text{2}}+{\text{B}}^{\text{2}}+2\text{AB}\cos \theta }=\text{C}$

$\text{C}=\sqrt{{\text{A}}^{\text{2}}+{\text{B}}^{\text{2}}+2\text{AB}\cos 120°}$

$\text{C}=\sqrt{{\text{A}}^{\text{2}}+{\text{B}}^{\text{2}}+2\text{AB}\left(-\frac{1}{2}\right)}$

$\Rightarrow \text{C}=\sqrt{{\text{A}}^{\text{2}}+{\text{B}}^{\text{2}}-\text{AB}}$

$\left|\overrightarrow{A}-\overrightarrow{B}\right|=\sqrt{{\text{A}}^{\text{2}}+{\text{B}}^{\text{2}}-2\text{AB}\cos 120°}$

$=\sqrt{{\text{A}}^{\text{2}}+{\text{B}}^{\text{2}}-2\text{AB}\left(-\frac{1}{2}\right)}$

$=\sqrt{{\text{A}}^{\text{2}}+{\text{B}}^{\text{2}}+\text{AB}}$