Here's some variations on MeowChow's idea above:
a perfectly respactable program, about 140 chars (after the #!.../):

#!/usr/local/bin/perl -ln0777

$\ ='
';
$c = 0;

s/^\s+//;
s/\s+/ /gs;

while ( s/(.{$c,}?. |.+$)// ) {

  print " " x (40 - ($c = 1 + length $1) / 2), $1
}
[download]

> triangle.pl a.txt w.txt

                                      a 
                                    a a a 
                                  a a a a a 
                                a a a a a a a 
                              a a a a a a a a a 
                             a a a a a a a a a a 
                                     The 
                                same principle 
                             applies to any text 
                          file, basically splitting 
                      words only on spaces or new lines
[download]

a one-liner, 74 chars (+6 extra flag chars):

> perl -ln0777we '$c=0;s/\s+/ /sg;print" "x(40-($c=length$1)/2),$1 whi
+le s/(.{$c,}?. |.+$)//' a.txt w.txt
                                       a 
                                      a a 
                                     a a a 
                                    a a a a 
                                   a a a a a 
                                  a a a a a a 
                                 a a a a a a a 
                                 a a a a a a a 
                                      The 
                                same principle 
                              applies to any text 
                           file, basically splitting 
                          words only on spaces or new 
                                     lines
[download]

and pretty well golfed, 66 chars (+6 extra flag chars):

> perl -ln0777e's/\s+/ /sg;print$"x(40-($c=length$&)/2),$&while s/.{0$
+c,}?. |.+$//' a.txt w.txt
                                       a 
                                      a a 
                                     a a a 
                                    a a a a 
                                   a a a a a 
                                  a a a a a a 
                                 a a a a a a a 
                                 a a a a a a a 
                               The same principle 
                            applies to any text file, 
                         basically splitting words only 
                             on spaces or new lines 

> perl -ln0777e's/\s+/ /sg;print$"x(40-($c=length$&)/2),$&while s/.{0$
+c,}?. |.+$//' w.txt a.txt

                                      The 
                                same principle 
                              applies to any text 
                           file, basically splitting 
                          words only on spaces or new 
                                     lines 
                                    a a a a 
                                   a a a a a 
                                  a a a a a a 
                                 a a a a a a a 
                                a a a a a a a a 
                                   a a a a a
[download]

p

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