Hi. I'm studying collision detection by the book "Real Time Collision Detection"

And I have a question about the math of checking if a point lies in a circle through 3d points in 2D.

Actually, I understand the principle of the equation or algorithm. However, the determinant expression 

is hard to understand for me, especially converting the second expression to third expression.

I know the row replacement in the determinant doesn't change the value of the determinant. So, I have a grasp of converting the first expression to second expression.

However, I can't tell the second conversion, on the third column.

How is it possible 

I will appreciate you ahead if you let me know this.

You can check this equation in the wikipedia https://en.wikipedia.org/wiki/Delaunay_triangulation

When computing a determinant, you can scale a column and add it to another column without changing the result. So if you add (-2D_x) times the first column and (-2D_y) times the second column to the third column, you'll get the transformation you see.

Thanks @alvaro !!

I didn't know why i didn't try to do a column replacement. 

Thanks to you, I can sleep well tonight.