After reading the article, I am aware of all the ways
that Monge's Theorem can be generalized. First, Monge's
Theorem is very similar to Desargues' Theorem.
Desargues' Theorem applies to two triangles and the
intersections of the lines through corresponding sides
of the triangles. Since we have discussed Monge's Theorem
in relation to dilations, this all seems to fit quite nicely.
The other very interesting generalization is of Monge's Theorem
to three dimensional objects. In this case, you would
consider */the lines* of intersection of tangent
planes to spheres!! Interesting! They also addresesed
the fact that the three circles of Monge's Theorem do
not have to be disjoint. The circles are allowed to intersect
or lie within one another. This would be an interesting thing
to investigate using a Sketchpad sketch. Finally, you can also
find a "collinear triple" using one external tangent
and two internal ones. These are considered "oriented tangents"