Subtracting Quarters of Squares Equals Multiply?! Can anyone explain to me how/why this works (hopefully in mostly layman's terms)?
It seems pretty magical to me at the moment.
$${{(a+b)^2\over4} - {(a-b)^2\over4}} = a  b.$$

In addition to the direct derivations already shown,
your magical equation is closely related to the formula
$$x^2 - y^2 = (x + y)(x - y).$$
Just set $a = x + y$ and $b = x - y.$
Then $\frac{a+b}2 = x$ and $\frac{a-b}2 = y,$
so  $x^2 = \frac{(a+b)^2}4$ and $y^2 = \frac{(a-b)^2}4.$
Use these facts to replace $x^2,\ y^2,\ x + y,$ and $x - y$
in the equation above and you will have derived your magical equation in $a$ and $b.$
