#!/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]

                                      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]

> 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]

> 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]

Very nice paring down, I especially like the conversion from a greedy expression to the minimal one you use, in order to avoid defining $c. Further reduction gives:

perl -ln0e's/\s+/ /g;print$"x(40-($c=length$&)/2),$&while/.{0$c,}? |.*
+/g' ...
[download]

61 chars of code plus 3 modifier chars.

   MeowChow                                   
               s aamecha.s a..a\u$&owag.print

Much cleaner! Putting the increment back in and not skipping a line at the end yields a "more intresting"* result (63 chars):

perl -ln0e's/\s+/ /g;print$"x(40-($c=2+length$&)/2),$&while/.{0$c,}? |
+.+/g' 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 -ln0e's/\s+/ /g;print$"x(40-($c=2+length$&)/2),$&while/.{0$c,}? |
+.+/g' 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]

(I know, I know. I'm the only one who could possibly still be interested at this point.)

* update "more interesting" meaning more <|triangular|>  (than the original challenge).

And, moving even deeper into 'useless use of code in void context':

print($"x(40-($c=2+length$&)/2),$&),$c*=($c<80)while...
[download]

will make triangles of files of any length (a step towards automated obfusification?).

p