What kind of pair is it?

Each line is written as $ax + by = c$. You will sort pairs of lines into the three cases without drawing them. Cross-multiplying the ratios avoids dividing by zero.

The idea. $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2}$ is the same as $a_1 b_2 = a_2 b_1$. If that fails, the lines cross once. If it holds, they are one line when $a_1 c_2 = a_2 c_1$ and $b_1 c_2 = b_2 c_1$ as well, and parallel otherwise.

Your tasks.

1. In crosses, return True when $a_1 b_2 \neq a_2 b_1$.
2. In kind, return "one", "none" or "many".
3. Fill kinds with the kind of each pair in pairs, in order.