Two circles with centers at (x_i, y_i) with radii r_i for i = 1, 2 are mutually tangent if (x_1 - x_2)^2 + (y_1 - y_2)^2 = (r_1 ± r_2)^2. If the center of the second circle is inside the first, then the - and + signs both correspond to internally tangent circles. If the center of the second circle is outside the first, then the - sign corresponds to externally tangent circles and the + sign to internally tangent circles. Finding the circles tangent to three given circles is known as Apollonius' problem. The Desborough Mirror, a beautiful bronze mirror made during the Iron Age between 50 BC and 50 AD, consists of arcs of circles that are exactly tangent.

Apollonius' problem | Casey's theorem | chain of circles | circle-circle tangents | circle packing | Descartes circle theorem | excircles | four coins problem | Hawaiian earring | incircle | inner Soddy circle | lens | lune | Malfatti circles | Malfatti's problem | nine circles theorem | outer Soddy circle | oval | Pappus chain | seven circles theorem | six circles theorem | Soddy circles | tangent curves | tangent spheres