Three circles

Given are three circles. For each pair of circles two tangent lines are drawn as shown below. (Three colors for three pairs). Each pair intersects at a point (see A, B and C).

It turns out that these the 3 pts of intersection are colinear! (see the thick green line).

Can you prove it? (Recall the hint -- consider the three-dimensional analoq of this construction)

The applet allows you to move the centers of the circles and the points A, B and C.

Created by Sumant, April 14, 2006, with GeoGebra