Common Tangents of Two Circles
What is a Tangent Line?
A tangent line to a circle is a line that touches the circle at exactly one point. This point is called the point of tangency.
A key property: the tangent line is always perpendicular to the radius drawn to the point of tangency. That is, if a tangent touches a circle at point $P$, and $O$ is the center, then $OP \perp \text{tangent line}$.
Common Tangents of Two Circles
A common tangent is a line that is tangent to two circles simultaneously.
There are two types of common tangents:
External Common Tangent (ECT)
A tangent line that does not pass between the two circles. Both circles lie on the same side of the tangent line.
Internal Common Tangent (ICT)
A tangent line that passes between the two circles. The circles lie on opposite sides of the tangent line.