Answer
$c$ bisects the angle between $a$ and $b$.
Work Step by Step
Consider $ \alpha$ to be the angle between $c$ and $a$. Then:
$cos \alpha =\dfrac{c \cdot a}{|c| |a|}$
$=\dfrac{(|a|b+|b|a) \cdot a}{|c| |a|}$
$=\dfrac{|a|b \cdot a+|b|a \cdot a}{|c| |a|}$
$=\dfrac{|a|b \cdot a+|b||a|^2}{|c| |a|}$
$=\dfrac{b \cdot a+|b||a|}{|c|}$
Now, consider $ \beta$ to be the angle between $c$ and $b$. Then:
$cos \alpha =\dfrac{c \cdot b}{|c| |b|}$
$=\dfrac{(|a|b+|b|a) \cdot a}{|c| |b|}$
$=\dfrac{|a|b \cdot b+|b|a \cdot b}{|c| |b|}$
$=\dfrac{|a||b|^2+|b|a \cdot b}{|c| |b|}$
$=\dfrac{|a||b|+a \cdot b}{|c|}$
As we can see from the above calculations, $cos \alpha =cos \beta$ , thus, $\alpha = \beta$
This shows that $c$ bisects the angle between $a$ and $b$.
