How about a regex?

#!/usr/bin/perl -l

"AB~ACFI~ADGJ~AE~BCDE~BFHJ~BI~EGHI~EJ~IJ" =~ /([^~])[^~]*([^~]).*~[^~]
+*([^~])[^~]*([^~])(?{local$z=$1 and local$y=$2 and local$x=$1 eq$3?$4
+:$1 eq$4?$3:($z=$2)&&($y=$1)&&$2 eq$3?$4:$2 eq$4?$3:0}).*~[^~]*((??{$
+y})[^~]*(??{$x})|(??{$x})[^~]*(??{$y}))(?{$x{join" - ",sort$x,$y,$z}+
++})(?!)/;
print for sort(keys %x), keys(%x) . " triangles found";
[download]

Sure it's overkill but it was fun =P

Only tested on perl 5.8.7 on windows and freebsd.