Relative Positions of Two Circles and Common Tangents

So far every idea has concerned a single circle. It's also useful to know how two circles can sit relative to one another, since this governs how many common tangent lines they share — a line that is simultaneously tangent to both. Let $C_1,C_2$ be the centres and $r_1,r_2$ the radii of two circles, and write $d=C_1C_2$ for the distance between the centres. There are five distinct possibilities:

1. $d>r_1+r_2$ — the circles are entirely apart from each other, neither touching nor overlapping. There are 4 common tangents: two 'direct' common tangents (which don't pass between the circles) and two 'transverse' common tangents (which cross between the circles).
2. $d=r_1+r_2$ — the circles touch externally at a single point, which lies on the segment $C_1C_2$ itself, dividing it internally in the ratio $r_1:r_2$. There are 3 common tangents: the two direct tangents, plus one tangent at the point of contact (the two transverse tangents have merged into this single one).
3. $|r_1-r_2|<d<r_1+r_2$ — the circles genuinely intersect in two distinct points. Only the 2 direct common tangents survive; no transverse tangent is possible once the circles overlap.
4. $d=|r_1-r_2|$ — the circles touch internally (one circle just grazing the inside of the other), touching at a point that divides $C_1C_2$ externally in the ratio $r_1:r_2$. There is only 1 common tangent, at the point of contact.
5. $d<|r_1-r_2|$ — one circle lies entirely inside the other without touching it. There are 0 common tangents.

The two points where the direct and transverse common tangents (when they exist) cross each other are called the external centre of similitude and internal centre of similitude respectively; the external one divides $C_1C_2$ externally in the ratio $r_1:r_2$, and the internal one divides it internally in the same ratio $r_1:r_2$ — the same division ratios that appear in the external/internal point-of-contact cases above, which is not a coincidence: those points of contact are the corresponding centre of similitude when the circles are actually touching.

Worked Example 1 (classify and count tangents). How are the circles $x^2+y^2=9$ (centre $O(0,0)$, $r_1=3$) and $(x-8)^2+y^2=4$ (centre $(8,0)$, $r_2=2$) related?

$d=C_1C_2=8$; $r_1+r_2=5$. Since $8>5$, the circles are apart, so there are 4 common tangents. …