Proof for $\frac{b}{h}=\frac{c(c-1)}{k(k-1)}$, a combinatorial identity

Question

Imagine we have $b$ teachers and $c$ students in a school such that

1. each teacher teaches $k$ students,
2. every two students have $h$ similar teachers.

I'd like to prove that $$\frac{b}{h}\:=\:\frac{c(c-1)}{k(k-1)}\,.$$ I thought about it for a little bit but weirdly enough, I have no insights on this.
A hint, an answer, or both would be appreciated.

Answer 1

Adress to each teacher a pair of students he/she teaches (so he adresses ${k\choose 2}$ pairs) and vice versa, to each pair of students a teacher they are listening (so they adresses $h$ teachers).

Then by double counting we have: $$b\cdot {k\choose 2} = {c\choose 2}\cdot h$$

Can you finsih now?