How to prove geometrically that $\langle a-b,b-c\rangle=\frac{1}{2} (|a-c|^2-|a-b|^2-|c-b|^2)$

Question

It is straighforward to prove algebraically that if $a,b,c$ are points in the Euclidean plane, then $$\langle a-b,b-c\rangle=\frac{1}{2} (|a-c|^2-|a-b|^2-|c-b|^2).$$ However, actually this formula seems to have a very geometric flavour, i.e. a formula involving the area of the rectangle whose sides are $bc$ and the projection of the segment $ab$ onto $bc$ and the areas of the squares constructed on the sides $ab$, $bc$ and $ac$ respectively.

Does anyone see an elementary geometric proof of that formula?

Answer 1

See picture, replacing $C$ by your $b$, $A$ by $a$ and $B$ by $c$.

This is the law of cosines.

Another one: