The argument of the absolute value function is twice thesignedarea of the triangle. It is positive if (x1, y1), (x2, y2) and (x3, y3) are vertices listed in one direction (clockwise or counterclockwise, I don't remember) and negative if they are listed in the opposite direction.

Which proof do you know?

One general way to derive this formula is to consider the signed area of the parallelogram built on vectors

and (*)

where , are the vectors of an orthonormal positively oriented basis. The signed area function takes two vectors and and returns a number. It has the following properties.

(i.e., the function is linear in both arguments)

If you expand (*) according to linearity, you should get the required formula.

For more, see the shoelace formula in Wikipedia.