(This picture is from the book Math Made Visual by Claudi Alsina and Roger B. Nelsen)
Given any two circles, there are exactly two lines that are tangent to both those circles at the same time. If the circles are different sizes, then these two tangents cross at some point.
If we have three circles, we can build three such points by looking at the common tangents to every pair of circles.
Monge’s theorem states that these three intersections always lie on a line.
The proof in the book is cute:
Let each circle be the “equator” of a sphere. Given a pair of spheres consider the cone generated by the two corresponding tangent lines. Half of the cone will lie above the plane of the circles and half will lie below. Now consider a plane tangent to the three half-spheres. This plane will also be tangent to each of the three cones, and it will intersect the original plane in a line L. Since this plane contains one line from each half-cone, the vertices of the three cones must be located on the intersection line L.