$\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a}\cdot \vec{c})\vec{b} - (\vec{a}\cdot \vec{b})\vec{c}$ 

Given equals $\dfrac{\vec{b} + \vec{c}}{\sqrt{2}}$ 

So compare coefficients: 

 $\vec{b}$ coefficient: 

$a\cdot c = \dfrac{1}{\sqrt{2}}$ $\vec{c}$ coefficient: $-(a\cdot b) = \dfrac{1}{\sqrt{2}}$ $\Rightarrow a\cdot b = -\dfrac{1}{\sqrt{2}}$ Thus angle $\theta$ between $a$ and $b$ satisfies: $\cos\theta = -\dfrac{1}{\sqrt{2}}$ $\Rightarrow \theta = \dfrac{3\pi}{4}$