Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. Prove that $$\mid G:N \mid = \sum \chi(1)^2$$ where the sum ranges over all irreducible characters of $G$ such that $N \subseteq \ker \chi$.
From previous results, I have proven that $N= \cap \ker \chi$, where the intersection goes over the same irreducible characters as above. I am trying to use this, but I don't manage. I am looking for hints.
The characters in the sum on the right hand side have a special relationship to $G/N$.