Definition: Given a fixed circle w with centre O and radius r the inverse of a point P (different from O) is the point P' such that OP.OP'=r^2 From the definition it follows that P and P' are inverses os each other points inside os the circle o inversion w transforms to points outside (and vice versa), and points on w invert to themselves. a) Any line through O, inverts to a circle a' through O (minus the point O itself) and vice versa.

Let $D$ be the common point of the two circles. Let $l$ be one of the common external tangents. The circle we are searching for is the inverse of the line $l$ w.r.t. the circle $(P,PD)$. We obtain one more circle if we use the other external common tangent.

http://www.sohu.com/a/162132709_614593 But it's Chinese.