Counting common tangents

You will write three functions that tell how two circles lie, count their common tangents and find the lengths of their common tangents. The checker uses the circles of the lesson.

The idea. Let $d$ be the distance of the centres. The number of common tangents is $4$, $3$, $2$, $1$ or $0$ as $d$ is more than $r_1 + r_2$, equal to it, between $|r_1 - r_2|$ and $r_1 + r_2$, equal to $|r_1 - r_2|$, or less than it. A direct tangent has length $\sqrt{d^2 - (r_1 - r_2)^2}$ and a transverse tangent has length $\sqrt{d^2 - (r_1 + r_2)^2}$.

Your tasks.

1. Finish count_tangents(d, r1, r2). Return the number of common tangents. Compare with a small tolerance, because the numbers are decimals.
2. Finish direct_length(d, r1, r2). Return the length of a direct tangent, or None if there is none.
3. Finish transverse_length(d, r1, r2). Return the length of a transverse tangent, or None if there is none.