This may, at least, get you started:

use strict;
use warnings;

my $valuesFile = 'file.txt';
my (@points, $count);
my @keys = qw(A X Y Z);

open FH, $valuesFile or die "Can't open $valuesFile: $!";

while(<FH>){
    chomp;
    my %pointHash;
    $pointHash{$keys[$count++]} = $_ for (split)[0, 2..4];
    push @points, \%pointHash;
    $count=0;
}

close FH or die "Can't close $valuesFile: $!";

my @results;

for my $x(0..$#points){
   my $p1 = $points[$x];
   for my $y (0..$#points){
        my $p2 = $points[$y];
        my $dist = sqrt(abs(( $p1->{X} - $p2->{X} ) **2
                 + ( $p1->{Y} - $p2->{Y} ) **2
                 + ( $p1->{Z} - $p2->{Z} ) **2)) ;  
        $results[$x][$y] = sprintf("%.0f", $dist);
   }
 }
#-------------------

print "\t";
print "$points[$_]->{A}\t" for(0..$#points);
print "\n";

$count = 0;

for my $r (@results){
    print "$points[$count++]->{A}\t";
    printf "%.0f\t",$_ for @$r;
    print "\n";
 }
[download]

Output:

    A    B    C    
A    0    4    5    
B    4    0    4    
C    5    4    0
[download]

Hope this helps!