# Understand the problem

Let $0 < a, b, c < 1$ with $ab + bc + ca = 1$. Prove that $$\frac{a}{1-a^2} + \frac{b}{1-b^2} + \frac{c}{1-c^2} \geq \frac {3 \sqrt{3}}{2}.$$

Determine when equality holds.

##### Source of the problem
Singapore Team Selection Test 2004
Inequalities
Medium
##### Suggested Book
Inequalities by BJ Venkatachala

Do you really need a hint? Try it first!

Show that there exists a triangle $\Delta ABC$ such that $a=\tan\frac{A}{2}, b=\tan\frac{B}{2}$ and $c=\tan\frac{C}{2}$. The easiest way to prove is is to define $A:=2\arctan a$, $B:=2\arctan b$ and show that $\tan\frac{\pi-A-B}{2}$ must be $c$.

Now the inequality becomes equivalent to $\tan A+\tan B+\tan C\ge 3\sqrt{3}$. Take a look at this well-known inequality. Note that the tangent is a convex function in $(0,\pi/2)$.

Note that, $a,b,c$ are less than $1$. This means that $\frac{A}{2},\frac{B}{2}$ and $\frac{C}{2}$ are all less than $\frac{\pi}{4}$. Hence the triangle is acute.

Using Jensen’s inequality, we get $\frac{\tan A+\tan B+\tan C}{3}\ge\tan\frac{A+B+C}{3}=\tan\frac{\pi}{3}=\sqrt{3}$. Equality holds for an equilateral triangle.