Consider a triangle ABC having the vertices $\mathrm { A } ( 1,2 ) , \mathrm { B } ( \alpha , \beta )$ and $\mathrm { C } ( \gamma , \delta )$ and angles $\angle A B C = \frac { \pi } { 6 }$ and $\angle B A C = \frac { 2 \pi } { 3 }$. If the points B and C lie on the line $y = x + 4$, then $\alpha ^ { 2 } + \gamma ^ { 2 }$ is equal to $\_\_\_\_$