<s>[INST] Prove that $\cos(2a) + \cos(2b) + \cos(2c) \geq -\frac{3}{2}$ for angles of a triangle Let the three internal angles of a triangle are $a,b,c$. Prove that 
$$\cos(2a) + \cos(2b) + \cos(2c) \geq -\frac{3}{2}.$$
I'm looking for an elementary, geometric proof. So avoid derivatives and tools from optimalization theory if it's possible.
 [/INST] If you are interested in an algebraic proof, mine goes as follows:
$\cos{2A} + \cos{2B} = 2 \cos{(A+B)} \cos{(A-B)} = %
2 \cos{(\pi - C)} \cos{(A-B)} = \\-2 \cos{C} \cos{(A-B)}$
Hence,
\begin{eqnarray*}
& \cos{2A} + \cos{2B} + \cos{2C} &\geq& - \dfrac{3}{2} \\
\iff & -2 \cos{C} \cos{(A-B)} + \cos{2C} &\geq& - \dfrac{3}{2} \\
\iff & -2 \cos{C} \cos{(A-B)} + 2\cos^2{C} - 1 &\geq& - \dfrac{3}{2} \\
\iff & 4\cos^2{C} - 4\cos{(A-B)} \cos{C} + 1 &\geq& 0
\end{eqnarray*}
We prove the last inequality with a quadratic function. 
Consider $f(t) = 4t^2 - 4 \cos{\phi} \; t + 1$, $t, \phi \in \mathbb{R}$. The descriminant is given by $D = 16(\cos^2{\phi} - 1) \leq 0$. Hence $f(t) \geq 0$ for all $t, \phi \in \mathbb{R}$. The result is now obvious if we let $t = \cos{C}$ and $\phi = A - B$.
Also, $f(t) = 0$ if and only if $\phi = n\pi$, $n \in \mathbb{Z}$, which is translated to $A = B$ for triangle $ABC$. In this case, $t = 1/2$, which means that $\cos{C} = 1/2 \implies C = \frac{\pi}{3}$. Thus, equality holds if and only if $A = B = C = \frac{\pi}{3}$. i.e. if and only if the triangle is equilateral.
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