Internally tangent circles occur when two circles on the same plane intersect at exactly one point and one circle lies within the other. Externally tangent circles occur when two circles on the same plane intersect at exactly one point, but one circle lies outside the other circle.
Circles are internally tangent if the distance between the centers of the circles is equal to the absolute value of the radius of the first circle minus the radius of the second circle. Internally tangent circles have a common internal tangent. A line that is tangent to a circle exists on the same plane as the circle and intersects the circle at exactly one point. A common internal tangent is a line tangent to two circles that also intersects the line that joins the center of the two circles. A common external tangent is a line tangent to two circles, but it does not intersect the line that joins the center of the two circles.
In mathematics, students may be asked to draw a circle that is tangent to three existing circles. This is known as Apollonius' Problem. In history, the Desborough Mirror is a bronze mirror composed of arcs of tangent circles. It was created somewhere between 50 B.C. and A.D. 50 during the Iron Age.