geometry - Cosine sum problem

Assume $sum_{i=1}^n a_i x_i=0$ with $x_i$ on the circle and $a_i>0$, $sum a_i=1$.
Then
$sum_{i=1}^nsum_{j=1}^n a_ilangle x_i,x_jrangle=0$.

It follows that for some $i$, $sum_{j=1}^nlangle x_i,x_jranglele 0$.

Since $langle x_i,x_irangle=1$, it implies
$sum_{j; jnot=i} langle x_i,x_jranglele -1$ which is what you want.