Triangle ABCABCABC has integer side lengths. Rectangles BCDE,ACFG,ABHJBCDE, ACFG, ABHJBCDE,ACFG,ABHJ are constructed so that CD=AC+ABCD = AC + ABCD=AC+AB, CF=AB+BCCF = AB + BCCF=AB+BC, and BH=(AC+BC)2BH = (AC + BC)^2BH=(AC+BC)2. If [ABHJ]=[BCDE]+[ACFG][ABHJ] = [BCDE] + [ACFG][ABHJ]=[BCDE]+[ACFG], how many different values can [ABC][ABC][ABC] have?
Details and assumptions
[PQRS][PQRS][PQRS] refers to the area of figure PQRSPQRSPQRS.