Relative Positions of Two Circles (Extra Information)

Relative Positions of Two Circles (Extra Information)

Given two circles, with centres $C_1,C_2$ and radii $r_1,r_2$, comparing the distance $d(C_1C_2)$ between their centres against $r_1$ and $r_2$ tells you exactly how the two circles sit relative to each other, and how many lines can be drawn tangent to both circles at once (a common tangent):

1. Touching externally: $d(C_1C_2)=r_1+r_2$. The circles meet at exactly one point, each lying outside the other everywhere else. Three common tangents can be drawn — two "direct" ones that don't cross between the circles, and one right at the point of contact.
2. Touching internally: $d(C_1C_2)=|r_1-r_2|$. One circle lies inside the other, touching it at exactly one point. Only one common tangent can be drawn, at the point of contact.
3. Disjoint (separate, neither inside the other): $r_1+r_2<d(C_1C_2)$. The circles don't meet at all. Four common tangents can be drawn — two direct tangents (not crossing between the circles) and two transverse tangents (crossing the segment joining the centres).
4. Intersecting in two points: the circles overlap and cross at two distinct points $A$ and $B$. The line through $A$ and $B$ is called the common chord, also known as the radical axis of the two circles. Exactly two common tangents can be drawn.
5. Concentric circles: the two circles share the same centre but have different radii, so one sits entirely inside the other with no point of contact at all. No common tangent can be drawn. …