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.

We obtain
$$\frac12 b\frac{ad-bc}{d-b}-\frac12 d\frac{ad-bc}{d-b}+\frac12d=\frac12 (b-d)\frac{ad-bc}{d-b}+\frac12d=\frac12(bc-ad+d)$$
