ELMO SL 2013 A4

Positive reals $a$ , $b$ , and $c$ obey $\frac{a^2+b^2+c^2}{ab+bc+ca} = \frac{ab+bc+ca+1}{2}$ . Prove that $$\sqrt{a^2+b^2+c^2} \le 1 + \frac{\lvert a-b \rvert + \lvert b-c \rvert + \lvert c-a \rvert}{2}.$$Proposed by Evan Chen