Two triangles are point perspective, if and only if, they are line perspective.
For a proof read my joint paper with Mary Garner Problemsolving and proving via generalization
With Desargues' theorem, one can easily prove the concurrency of the medians as shown at Concurrency of medians with Desargues
The theorem is also applied several times in Michael Fox's proof of the BMO-Brianchon corollary at A British Math Olympiad problem and its dual
Desargues' Theorem also beautifully displays the projective duality between "points" and "lines", and between "collinearity" and "concurrency". Do a Google search to read up more on "projective duality".
For a poem of mine written many years ago related to the topic of duality, go to Duality
Last modified, Michael de Villiers, 31 October 2010.