[P.S.: Determinant III does actually _factor_ as (ab'-a'b)^2, once I set c=-a-b. Note that ab'-a'b is _pure imaginary_, so its square is a negative real number. I usually denote ab'-a'b as -2i(axb), where "axb" is my "cross product" of complex numbers. So Determinant III = (-2i(axb))^2=-4(axb)^2. We now have 2 different ways of factoring the same expression: -4(axb)^2 = -(-|c|+|b|+|a|)(|c|-|b|+|a|)(|c|+|b|-|a|)(|c|+|b|+|a|) This isn't a problem unless a,b are Gaussian integers and |a|,|b|,|c| are real integers; in this case, we have to be able to pair up the factors on the right hand side to make a square integer. Is there a name for triangles whose sides are integers, and _which can also be embedded in the plane with integer coordinates for the vertices_ ?? ] The Cayley-Menger determinant form for Heron's formula of the area of a triangle is well known: http://mathworld.wolfram.com/HeronsFormula.html -16 (Triangle Area)^2 = (Determinant "I") |0 a b c| |a 0 c b| |b c 0 a| = |c b a 0| (Determinant "II") |0 1 1 1| |1 0 cc bb| |1 cc 0 aa| = |1 bb aa 0| (c - b - a) (c - b + a) (c + b - a) (c + b + a) = (|c| - |b| - |a|) (|c| - |b| + |a|) (|c| + |b| - |a|) (|c| + |b| + |a|) where a,b,c are the (real) triangle side lengths, and aa=a*a=|a|^2, bb=b*b=|b|^2, cc=c*c=|c|^2. Determinant II is obtained from Determinant I by pre & post multiplication by real diagonal matrices, and vice versa. E.g., see Muir's Theory of Determinants, p.44. I know that determinantal identities are a dime a dozen, but here's one which I believe is new: This time, instead of using a,b,c for the (real) triangle side lengths, I use a,b,c for the _complex_ triangle side _vectors_, arranged head-to-tail so that a+b+c=0, and a',b',c' are the complex conjugates of a,b,c, respectively. Then, -16 (Triangle Area)^2 = (Determinant "III") |0 a b c | |a 0 c' b'| |b c' 0 a'| |c b' a' 0 | (Determinant "IV") |0 1 1 1 | |1 0 cc' bb'| |1 cc' 0 aa'| |1 bb' aa' 0 | Note that Determinant IV is _real_ and is _identical_ to Determinant II, because aa'=|a|^2, bb'=|b|^2, cc'=|c|^2. Determinants III and IV are interconvertible using pre & post multiplication by diagonal matrices with _complex_ entries. We also note that while Determinant IV = II = I can be factored, Determinant III cannot, because |a|=sqrt(aa'), |b|=sqrt(bb'), |c|=sqrt(cc') are not necessarily rational. Matrix III is also a rare example of a _symmetric_ complex matrix that is _not Hermitian_.