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If $\vec{a},\vec{b},\vec{c}$ are coplanar vectors and $\vec{a}$ is not parallel to $\vec{b}$, then prove that $\{{(\vec{c} \times \vec{b}) \cdot(\vec{a} \times \vec{b})} \}\vec{a}+\{{(\vec{a} \times \vec{c}) \cdot(\vec{a} \times \vec{b})} \}\vec{b}$ is equal to $\{{(\vec{a} \times \vec{b}) \cdot(\vec{a} \times \vec{b})} \}\vec{c}$

I am trying to use the fact that three coplanar vectors are linearly dependent but not able to get the desired result. Could someone please help me with this.

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    If $\vec a$ and $\vec b$ are not parallel, then $\vec c=x\vec a+y\vec b$. You could try inserting that and distribute as much as possible. – 2017-02-02

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I am trying to use the fact that three coplanar vectors are linearly dependent

Hint: that's a good idea. Let $\vec c=\lambda \vec a+\mu \vec b$ then distributing and using $\vec u \cdot \vec u = |\vec u|^2\,$, $\vec u \times \vec u = 0\,$:

$$\lambda |\vec a\times \vec b|^2 \;\vec a + \mu |\vec a \times \vec b|^2 \;\vec b = |\vec a \times \vec b|^2\,(\lambda \vec a + \mu \vec b) = |\vec a \times \vec b|^2\;\vec c$$