Tangencies: Three Tangent Circles

Any three points can be the centers of three mutually tangent circles. To construct the circles, form a triangle from the three centers, bisect its angles (blue), and drop perpendiculars from the point where the bisectors meet to the three sides (green). The points where these perpendiculars cross the sides are the desired points of tangency. They are also the points where an inscribed circle (red) is tangent to the triangle; this circle has its center at the point where the six lines meet, and crosses the three tangent circles perpendicularly at their tangent points.

The same construction works to form two circles inside a third, but you should use two external bisectors and one internal bisector instead of three internal bisectors. (Curious fact: three bisectors meet iff the number of external ones is even.)

Similarly to this pattern, four tangent circles also always have a circle through their tangencies, but it is not always perpendicular to them.