Properties of self-dual code Let $C $ be a self-dual binary $[n,k,d]$ code.

I want to show that if $ c=(c_1, \dots, c_n) \in C $ then $ \sum_{i=1}^n c_i=0$ and that all the words of the code have an even weight. 

We know that $ x \in C^{\perp} \Leftrightarrow Gx=0$ where $ G $ is the generator matrix.

Since $ C=C^{\perp}$ we have that  $ G=H $ where $ H$ is the parity matrix of the code. 

So we have that $ c=(c_1, \dots, c_n) \in C \Rightarrow c \in C^{\perp} \Rightarrow Gx=0 \Rightarrow Hx=0 $

Do we get from that,  that $ \sum_{i=1}^n c_i=0$?

If so how?
By definition, the dual code $C^\perp$ consists of all words $x$ such that $\langle x,y \rangle = 0$ for all $y \in C$. If $C = C^\perp$ then in particular $\langle x,x \rangle = 0$ for all $x \in C$, which implies that $x$ has even weight.