Deasargues' Theorem asserts:
If we are given generic triples of points A, B, C and A', B', C' in
the plane, then the intersections AB.A'B', AC.A'C', and BC.B'C'
all lie on one line.
The simplest way to see this result is to visualize the
figure as the projection of one (say with the same labels) in 3D.
In this case the assertion
is clear, since the line is the intersection of the two 3D planes
spanned by the points A, B, C and A', B', C'.
In this figure A, B, C are the vertices of one triangle,
and A', B', C' are the vertices of the other.