Desargues' Theorem
Desargues' Theorem: Two triangles are in perspective axially if and only if they are in perspective centrally.
In terms of the picture provided, lines AD, BE, and CF intersect in a common point G if and only if the points H, I, J lie on a common line.
For this applet, the Line of Perspectivity (on its own and via Y & Z), A, B, D, E, H, I, J all manipulate; H, I, J are animated. The Point of Perspectivity (G) shows the intersection of AD, BE, and CF as a result of the other points' locations.