If $G$ is the centroid, then: $GA \bot GB \Leftrightarrow {c^2} = \frac{4}{9}\left( {{m_a}^2 + {m_b}^2} \right) = \frac{4}{9}\left( {\frac{{2{b^2} + 2{c^2} - {a^2}}}{4} + \frac{{2{a^2} + 2{c^2} - {b^2}}}{4}} \right)$ ${c^2} = \frac{{4{c^2} + {a^2} + {b^2}}}{9} \Leftrightarrow $ $\boxed{a^2+b^2=5c^2}$