THEOREM: The "Heronian" triangles can also be characterized as the triangles with integer sides and rational area (they then automatically have integer area). CORRECTED PROOF: For a primitive triangle, with sides a,b,c integer with GCD(a,b,c)=1, (a,b,c)=(even,even,even) is impossible (not primitive). Consider a,b,c mod 4. The squares mod 4 are 0 and 1. The only ways 16*area^2=p*(p-2a)*(p-2b)*(p-2c) where p=a+b+c can be a square, as we find by exhaustive consideration of all 4*4*4=64 possibilities [then exclude cases with all of {a,b,c} even], are these six: (a,b,c) mod 4 = (0,1,1) (0,1,3) (0,3,3) (1,1,2) (1,2,3) (2,3,3) [or some re-order of those]. in which cases 16*area^2 is found to be 0 mod 64 in every case. Hence area is an integer (not a quarter integer). QED. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)