Three rigidly-linked particles in space have the $6$ degrees of freedom of a rigid body

A dynamical system consists of three particles $m_1,m_2,m_3$ in motion in space. The distance between $m_1$ and $m_2$, $m_2$ and $m_3$, $m_1$ and $m_3$ remain constant throughout the motion. The motion will be governed by $n$ Euler Lagrange equations. Then: