How to prove that $ \sin \angle{GAB}+\sin \angle{GBC}+\sin \angle{GCA} \le \frac{3}{2} $ for a triangle $ABC$ with centroid $G$? Let $ G $ be the centroid of $ \triangle ABC $ , such that  $ \measuredangle{GAB}=x,\measuredangle{GBC}=y,\measuredangle{GCA}=z $.

How do I prove  that :
$$  \sin x +\sin y +\sin z\le \frac{3}{2} $$
With little effort, one can prove that $$\sin x = \frac{1}{\sqrt{1+(2\cot\alpha+\cot\beta)^2}}$$, where $\alpha = \angle A, \beta = \angle B, \gamma = \angle C$. It's also easy to check that the cotangents of angles of any triangle satisfies the following: $$\cot\alpha\cot\beta+\cot\beta\cot\gamma+\cot\gamma\cot\alpha = 1$$. Therefore, the problem is equivalent to the following inequality: 
$$x,y,z\in\mathbb{R} : xy+yz+zx=1\rightarrow\sum_{x,y,z}\sqrt\frac{1}{1+(2x+y)^2}\leq\frac{3}{2}$$ 
Now, while I believe that an avid contest-type inequality solver could find an elegant solution to above, it can also be solved by Lagrange multiplier method. 
To begin with, let's make the following substituion $$(2x+y, 2y+z, 2z+x)= (a,b,c)\leftrightarrow (x,y,z) = (\frac{4a-2b+c}{9},\frac{4b-2c+a}{9},\frac{4c-2a+b}{9})$$
Now our problem is equivalent to $$-2(a^2+b^2+c^2)+5(ab+bc+ca) = 27\rightarrow \sum_{a,b,c}\frac{1}{\sqrt {1+a^2}}\leq\frac{3}{2}$$.
Now it's a routine exercise of Lagrange multiplier method that yields the inequality is attained where $a=b=c=3x=3y=3z$, which means our triangle must be equilateral for this to hold.
There is a theorem in geometry telling the point $G$ is inside the triangle.

So , Join it to $A , B$ and $C.$

It is obvious that $x + y + z &lt; 90$

By using lagrange extreme theorem for multi variable functions, Find the maximum 

of $\sin x + \sin y + \sin z$ with the condition $x + y + z &lt; 90$

Its maximum is $\frac{3}{2}$