Common Tangents and Centres of similitude
i) (a) C1C2 > (r1 + r2) and r1 ≠ r2.
There are two pairs of common tangents:
Transverse pair ‐ the pair of common tangents (1 & 2) intersecting at a
point Q on the line segment 
.
Direct pair ‐ the pair of common tangents intersecting at a point P not on
.
The points P, Q, C1 and C2 lie on one straight line i.e, they are
collinear.
Straight line if
Point P is called internal centre of similitude
Point Q is called external centre of similitude
P divides C1C2 internally in the ratio r1 :
r2
Q divides C1C2 externally in the same ratio.
i) (b) C1C2 > (r1 + r2) and r1 = r2.
Tangents 1 & 2 and C1C2 are parallel. Hence they do
not meet.
Hence there is no external centre of similitude.
How to find the equations of parallel common tangents:
Let y = mx + c be the tangent equation.
The slope m = slope of C1C2 (∵ Parallel lines
have same slope).
The radius (r1 = r2) is the perpendicular distance.
r1 = 
From the above, 'c' can be found.
Knowing both 'm' and 'c', the tangent equation can be obtained.
ii) = r1 + r2
P is the contact of point of the two circles.
At 'P' there is only one common tangent.
P is still called the internal centre of similitude.
iii) |r1 – r2| <
< (r1 + r2)
Note that in this case, internal centre of similitude does not exist.
Length of common tangent
If d is the distance between centres of two circles whose radii are r1 and r2
then length of direct common tangent of two circles is 
and length of
transverse common tangent of two circles is
.
and length of
transverse common tangent of two circles is
.