Official solution
Let A(1, 2, 3), B\[\left( { - {7 \over 3}, - {4 \over 3}, - {1 \over 3}} \right)\]
\[ \therefore \] Midpoint of AB = M = \[\left( {{{{{ - 7} \over 3} + 1} \over 2},{{{{ - 4} \over 3} + 2} \over 2},{{{{ - 1} \over 3} + 3} \over 2}} \right)\]
= \[\left( {{{ - 2} \over 3},{1 \over 3},{4 \over 3}} \right)\]
DR of AM = \[\left( {1 + {2 \over 3},2 - {1 \over 3},3 - {4 \over 3}} \right)\]
= \[\left( {{5 \over 3},{5 \over 3},{5 \over 3}} \right)\]
= (1, 1, 1)
Equation of plane
\[a\left( {x + {2 \over 3}} \right) + b\left( {y - {1 \over 3}} \right) + c\left( {z - {4 \over 3}} \right)\] = 0
\[ \Rightarrow \] \[1\left( {x + {2 \over 3}} \right) + 1\left( {y - {1 \over 3}} \right) + 1\left( {z - {4 \over 3}} \right)\] = 0
\[ \Rightarrow \] x + y + z = 1
\[ \therefore \] (1, --1, 1) lies on the plane.