Consider a sphere in $\Bbb{R^n}$ centered on $P$. Let $Q$ be a point where some tangent plane meets the sphere. In particular $Q$ is the point on the plane which is the closest to $P$. If $X$ is a point on this tangent plan then $(Q-P)(Q-X) = 0$, and so: $$(Q-P)X = Q^2-PQ.$$ Now let's isolate $Q^2$ in two different ways:
By Pythagoras: $Q^2= P^2-(P-Q)^2$.
$(P-Q)^2 = P^2+Q^2-PQ - QP$, so $$Q^2=(P-Q)^2-P^2+PQ+QP.$$
But these two are not the same. What am I doing wrong?
Pythagoras' theorem that tells you that $$Q^2+(P-Q)^2=P^2,$$ if $P$ and $Q$ are perpendicular, which need not be the case here. In general $$(P+Q)^2=P^2+Q^2+PQ+QP,\tag{1}$$ and in particular, if $P$ and $Q$ are perpendicular this shows that $$(P+Q)^2=P^2+Q^2.$$ Replacing $Q$ by $-Q$ in equation $(1)$ yields the identity used in the book: $$(P-Q)^2=P^2+Q^2-PQ-QP,$$ and if $P$ and $Q$ are perpendicular this yields your identity: $$(P-Q)^2=P^2+Q^2.$$