If A, B , C ,D are any four points in space, prove that `|vec(AB)xx vec(CD)xxvec(BC) xx vec(AD) + vec(CA) xx vec(BD)|`= 4 ( area of triangle ABC).

Explanation

Let the position vectors of points A,B,C,D be `veca, vecb, vecc and vecd`, respectively, with respect to some origin.
`|vec(AB)xxvec(CD)+vec(BC)xx vec(AD)+vec(CA)xx vec(BD)|`
`[|(vecb-veca)xx(vecd-vecc)+ (vecc-vecb) xx (vecd -veca) + (veca -vecc) xx (vecd-vecb)|`
`2|vecb xx veca +vecc xx vecb +vecaxxvecc|`
`2(2 xx ("area of " triangleABC)`
` 4xx (" area of " triangle ABC)`