8.3 Common Tangents

8.3 Common Tangents
A common tangent to two circles is a straight line that touches each of the circles at only one point.

1. Intersect at two points
(a) Circles of the same size
Number of common tangents
Properties of common tangents
Two common tangents:
AB and CD
AC = BD
AB = CD
AB parallel to // OR parallel to // CD



(b) Circles of different sizes


Number of common tangents
Properties of common tangents
Two common tangents:
ABE and CDE
AB = CD
BE = DE
OA // RB
OC // RD


2. Intersect at one point
(a) Circles of the same size

Number of common tangents
Properties of common tangents
Three common tangents:
AB, CD and PQ
AC = PQ = BD
AB = OR = CD
AB // OR // CD
AC // PQ // BD
PQ perpendicular to OR


(b) Circles of different sizes
(i) External contact


Number of common tangents
Properties of common tangents
Three common tangents:
ABE, CDE and PQ
AB = CD
BE = DE
OA // RB
OC // RD
PQ perpendicular to ORE


(ii) Internal contact


Number of common tangents
Properties of common tangents
One common tangent: PQR
ONQ perpendicular to PR


3. Do not intersect
(a) Circles of the same size

Number of common tangents
Properties of common tangents
Four common tangents:
AB, CD, PS and RQ
AB = CD = OV
PS = RQ
AB // OV // CD


(b) Circles of different sizes


Number of common tangents
Properties of common tangents
Four common tangents:
AB, CD, PS and RQ
AB = CD
BT = DT
PS = RQ
OA // VB
OC // VD