= Solution
<Momentum> conservation is $P_A+P_B=P_C+P_D$. The diagonal bilinears are $P_a^2=m_a^2c^2$. The six off-diagonal ones are
$$
2P_A\cdot P_B=s-(m_A^2+m_B^2)c^2,
$$
$$
2P_A\cdot P_C=(m_A^2+m_C^2)c^2-t,
\qquad
2P_A\cdot P_D=(m_A^2+m_D^2)c^2-u,
$$
$$
2P_B\cdot P_C=(m_B^2+m_C^2)c^2-u,
\qquad
2P_B\cdot P_D=(m_B^2+m_D^2)c^2-t,
$$
and
$$
2P_C\cdot P_D=s-(m_C^2+m_D^2)c^2.
$$
These express every bilinear invariant through the <Mandelstam variables> and masses. Expanding $s+t+u$ and using <momentum> conservation cancels all mixed products, leaving
$$
\boxed{s+t+u=(m_A^2+m_B^2+m_C^2+m_D^2)c^2}.
$$
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