Consider a triangle ABC having the vertices A (1,2), B (α, β) and C (γ, δ) and angles ∠ A B C=(π)/6 and ∠ B A C=(2 π)/3. If the points B and C lie on…

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