Assume that u,v,w are vectors in a inner product space $V$, then
$||w-1/2(u+v)^{2}||=1/2(||w-v||^{2}+||w+v||^{2})-1/4||u-v||^{2}$
Here is a solution, http://linearalgebras.com/6A.html Question 27. But I cannot find how the author manipulating the equation obtained by plugging in to find the answer. Could someone show the steps explicitly? Thanks in advance!
The solution you linked is incorrect. You should substitute $$ a = w-\frac12(u+v) $$ and $$ b = \frac12(u-v) $$ into the identity. Then simplify and rearrange.