<s>[INST] $M$ is a point in an equaliateral $ABC$ of area $S$. $S'$ is the area of the triangle with sides $MA,MB,MC$. Prove that $S'\leq \frac{1}{3}S$. 
$M$ is a point in an equilateral triangle $ABC$ with the area $S$. Prove that $MA, MB, MC$ are the lengths of three sides of a triangle which has area $$S'\leq \frac{1}{3}S$$

 [/INST] Choose a coordinate system so that triangle $ABC$ is lying on the unit circle centered at origin and $A$ on the $x$-axis. Let $a = AM$, $b = BM$, $c = CM$ and $S'$ be the area of a triangle with sides $a,b,c$. In this coordinate system, $S = \frac{3\sqrt{3}}{4}$, we want to show $S' \le \frac{\sqrt{3}}{4}$. Using Heron's formula, this is equivalent to
$$16S'^2 = (a^2+b^2+c^2)^2 - 2(a^4+b^4+c^4) \stackrel{?}{\le} 3$$
Identify euclidean plane with the complex plane. The vertices $A,B,C$ corresponds to $1, \omega, \omega^2 \in \mathbb{C}$ where $\omega = e^{\frac{2\pi}{3}i}$ is the cubic root of unity. Let $z$ be the complex number corresponds to $M$ and $\rho = |z|$, we have
$$
\begin{cases}
a^2 = |z-1|^2 = \rho^2 + 1  - (z + \bar{z})\\
b^2 = |z-\omega|^2 = \rho^2 + 1 - (z\omega + \bar{z}\omega^2)\\
c^2 = |z-\omega^2|^2 = \rho^2 + 1 - (z\omega^2 + \bar{z}\omega)
\end{cases}
\implies
a^2 + b^2 + c^2 = 3(\rho^2 + 1)
$$
Thanks to the identity $\omega^2 + \omega + 1 = 0$, all cross terms involving $\omega$ explicitly get canceled out.   
Doing the same thing to $a^4 + b^4 + c^4$, we get
$$\begin{align}a^4 + b^4 + c^4 
&= \sum_{k=0}^2 (\rho^2 + 1 + (z\omega^k + \bar{z}\omega^{-k}))^2\\
&= \sum_{k=0}^2\left[ (\rho^2 + 1)^2 + (z\omega^k + \bar{z}\omega^{-k})^2\right]\\
&= 3(\rho^2 + 1)^2 + 6\rho^2\end{align}$$
Combine these, we obtain
$$16S'^2 = 3(\rho^2+1)^2 - 12\rho^2 = 3(1 - \rho^2)^2$$
Since $M$ is inside triangle $ABC$, we have $\rho^2 \le 1$. As a result,
$$S' = \frac{\sqrt{3}}{4}(1-\rho^2) \le \frac{\sqrt{3}}{4} = \frac13 S$$
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