Given that the constraint of $a, b, c$, for which $a, b, c$ are non-negative real numbers, is $a^2+b^2+c^2=1,$ find the maximum value of $$\frac{a}{(1+bc)}+\frac b{(1+ac)}+\frac{c}{(1+ab)}.$$
For this question I have tried using this geometric method, hopefully it can be logically correct
Do you all have actually better method(s) to solve this problem?
For non-negative variables we obtain: $$\sum_{cyc}\frac{a}{1+bc}\leq\sum_{cyc}\frac{a\sqrt2}{a+b+c}=\sqrt2$$ because $$2(1+bc)^2\geq(a+b+c)^2$$ it's $$2+4bc+2b^2c^2\geq1+2(ab+ac+bc)$$ or $$2b^2c^2+(b+c-a)^2\geq0.$$ The equality occurs fot $c=0$ and $a=b=\frac{1}{\sqrt2},$ which says that we got a maximal value.