Determinant of 4x4 matrix equals determinant of 3x3 matrix... but how does this work?

Question

I found this 4x4 matrix on wikipedia. It's determinant is used to tell if a vector D in R2 lies inside, outside or on the circumcircle of a triangle ABC.

What I fail to understand is:

Why does the determinant of the 4x4 matrix equal the determinant of the simpler 3x3 matrix and why does it seem to make no difference if you say (A^2 - D^2) or (A - D)^2 in this case?

Thanks for your answers, I'm just beginning to learn linear algebra.

Have a nice day!

PS

this is the link to the wikipedia article https://en.wikipedia.org/wiki/Delaunay_triangulation

Answer 1

1. Subtracting one row from another does not change the determinant. Apply this rule three times to see that subtracting the last row from the others leaves a matrix whose last column has three $0$ and one $1$. Then apply Laplace expansion on the last column.

2. Subtract $2D_x$ times the first column from the third column. Subtract $2D_y$ times the second column from the third column.