Need help understanding statement "By linear algebra we know $\left|A,B,C\right|=-(A\times C)\cdot B=-(C\times B)\cdot A$

Question

I am reading a paper for a famous ray-triangle intersection procedure https://cadxfem.org/inf/Fast%20MinimumStorage%20RayTriangle%20Intersection.pdf

They use Cramer's rule to solve a set of equations but do the simplification described in the title:

$\left|A,B,C\right|=-(A\times C)\cdot B=-(C\times B)\cdot A$

I'm wondering how it is that these are equal.

Posting image from paper, here are some variable explanations:

1. $D$ = ray direction
2. $E1, E2$ = triangle edges
3. $T$ = translation of ray start to origin
4. $t$ = ray distance
5. $u,v$ = two of three barycentric coordinates

Answer 1

These are properties of the Triple product https://en.wikipedia.org/wiki/Triple_product

The first step is a property where the det(a,b,c) = dot(a, cross(b,c))

The second is a property of scalar product where dot(a, cross(b,c)) = dot(cross(a,b),c)