Problem
GEO-B3-M01-P018 Common Tangents via Concentric Circles
Two disjoint circles, neither inside the other, are given. Prove that there exists an inversion centered on their line of centers after which the circles become concentric, and explain how this helps construct their common tangents.
Hint 1. We need an inversion center from which tangent lengths to the given circles are equal.
Hint 2. Such a point is obtained from the radical axis and the line of centers.
E. Full solution. Choose point \(O\) on the line of centers so that its powers with respect to the two circles are equal. Then the tangent lengths from \(O\) to the two circles are equal. Take inversion centered at \(O\) with power equal to this common power. Under this inversion, both circles transform in a coordinated way into circles whose centers lie on the same line and become concentric. Then common tangents are constructed as inverse images of lines tangent to the two concentric circles, namely lines at a fixed distance from the common center.
The problem shows that the choice of inversion center is often dictated by powers with respect to circles.