[math-fun] Hero[n]'s formula

As all munsters know, area^2 = s(s-a)(s-b)(s-c) = xyz(x+y+z). Is there a really good proof of this? The best one I've been able to cook up[1] says: let t,u,v = tan A/2 etc.; then xt=r (the inradius) etc., so t:u:v = 1/x:1/y:1/z, and (tangent-sum formula, angle-sum of triangle) uv+vt+tu=1, so t = 1/x / sqrt(1/yz+1/zx+1/xy), so r^2 = 1/(1/yz+1/zx+1/xy) = xyz/(x+y+z) and as rs=area we're done. But this feels a bit too complicated. Is there perhaps a neat proof in 4 or 6 dimensions that looks at the cartesian product of the triangle (or some tetrahedron with the triangle as base) with itself? [1] Presumably it's been known for centuries. -- g