Vector identity in $\mathbb{R}^2$

Question

I was told there exists an identity of the following type: $$ \frac{A-B}{|A-B|^2}\cdot\frac{A-C}{|A-C|^2}+\frac{B-A}{|B-A|^2}\cdot\frac{B-C}{|B-C|^2}+\frac{C-A}{|C-A|^2}\cdot\frac{C-B}{|C-B|^2}=\frac{const}{R_{A,B,C}}, $$ where $A,B,C$ are vectors in $\mathbb{R}^2$ and $R_{A,B,C}$ is the radius of the circle passing through $A,B,C$. Can anybody confirm this and/or give me a reference for that?

Answer 1

WLOG the centre of the circle is the origin. Take $A = r [\cos(\alpha),\sin(\alpha)]$, $B = r [\cos(\beta), \sin(\beta)]$, $C = r [\cos(\gamma),\sin(\gamma)]$, so $R_{A,B,C} = r$. Then after some calculation the left side simplifies to $ 1/(2 r^2)$.