Simplification of an algebraic determinant After studying some analytic geometry, I came across this step in a solution, however, I am not how they managed to simplify the determinant in this way.

When I tried to evaluate this, I got:
$\frac{bc-ad}{2}+\frac{ad-bc}{2b-2d}$, but didn’t see how this got to the desired form. Many thanks.

Expand the first on the first line and the second on the second line. You obtain $$\frac12b\cdot\frac{ad-bc}{d-b}-\frac12\left(-d+d\cdot\frac{ad-bc}{d-b}\right)=\frac d2+(b-d)\frac{ad-bc}{2(d-b)}=\frac{d-ad+bc}{2}$$
