Monge-D'Alembert Circle Theorem

Before reading this post, please read this post.

The theorem:

"Given any three circles in a plane, none of which is inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinear"

Looks fancy and hard, but trust me, the theorem is trivial by [Desargues theorem][https://www.expii.com/e/1177/] ( in which can be proven by Menelaus theorem ).

So how to start? Lets find the "2 reference triangles" we need. So we "wish" that the 3 points are colinear, and we try to find the respective "3 pair of concurrent points".

Recall that the intersection of external tangent lines lie on the line passes through the two centers. So why not draw the three centers of the circles?