<s>[INST] Triangles geometric inequalities Is there any technique for geometric inequalities for triangles? For example, I tried to use Ravi’s substitution
$a=x+y, b=y+z, c=z+x, s=x+y+z, s-a=z, s-b=x, s-c=y,
A= \sqrt { xyz(x+y+z) }$
and
$r= \frac {A}{s} = \sqrt{\frac {xyz}{x+y+z}}),$
$R = \frac {abc} {4A}=\frac {(x+y)(y+z)(z+x) } {4 \sqrt { xyz(x+y+z) }}$
$\frac{r}{2R} = \frac {2xyz}{(x+y)(y+z)(z+x)} $
$r_a= \sqrt { \frac {xy}{z} (x+y+z)},$
to prove the inequality
$\sum_{cyc}^{} \frac {r_a}{a} \ge \sqrt {3(2+ \frac{r}{2R})}$
which becomes
$\sum_{cyc}^{} \sqrt {\frac {xy}{z} \frac{x+y+z}{x+y}} \ge  \sqrt {3 (2+ \frac {2xyz}{(x+y)(y+z)(z+x) })}$
Then I tried to prove that
$\sin \frac{A}{2} + \sin \frac{B}{2}+ \sin \frac{C}{2} \le \sqrt {2 + \frac{r}{2R}} = \sqrt {2+ \frac {2xyz}{(x+y)(y+z)(z+x) }}$
Since 
$\sin \frac{A}{2} + \sin \frac{B}{2}+ \sin \frac{C}{2} \le \frac {3}{2}$
then we have to prove
$\frac {3}{2} \le \sqrt {2+ \frac {2xyz}{(x+y)(y+z)(z+x) }}$
but I stuck. Thank you
 [/INST] A proof of $$\sin \frac{A}{2} + \sin \frac{B}{2}+ \sin \frac{C}{2} \le \sqrt {2 + \frac{r}{2R}}.$$
We need to prove that
$$\sum_{cyc}\sqrt{\frac{1-\frac{b^2+c^2-a^2}{2bc}}{2}}\leq\sqrt{2+\frac{\frac{2S}{a+b+c}}{\frac{abc}{2S}}}$$ or
$$\sum_{cyc}\sqrt{\frac{(a+b-c)(a+c-b)}{4bc}}\leq\sqrt{2+\frac{16S^2}{4(a+b+c)abc}}$$ or
$$\sum_{cyc}\sqrt{\frac{(a+b-c)(a+c-b)}{bc}}\leq\sqrt{\frac{8abc+(a+b-c)(a+c-b)(b+c-a)}{abc}}$$ or
$$\sum_{cyc}\sqrt{a(a+b-c)(a+c-b)}\leq\sqrt{8abc+(a+b-c)(a+c-b)(b+c-a)}$$ or
$$\sum_{cyc}\sqrt{(y+z)4yz}\leq\sqrt{8(x+y)(x+z)(y+z)+8xyz}$$ or
$$\sqrt{2(x+y+z)(xy+xz+yz)}\geq\sum_{cyc}\sqrt{xy(x+y)}$$ or
$$2\sum_{cyc}(x^2y+x^2z+xyz)\geq\sum_{cyc}(x^2y+x^2z+2\sqrt{xyxz(x+y)(x+z)})$$ or
$$\sum_{cyc}(x^2y+x^2z+2xyz)\geq2\sum_{cyc}x\sqrt{yz(x+y)(x+z)},$$
which is true by AM-GM:
$$2\sum_{cyc}x\sqrt{yz(x+y)(x+z)}\leq\sum_{cyc}x(y(x+z)+z(x+y))=\sum_{cyc}(x^2y+x^2z+2xyz).$$
Done!
A proof of 
$$\sum_{cyc}^{} \frac {r_a}{a} \ge \sqrt {3(2+ \frac{r}{2R})}.$$
We need to prove that
$$\sum_{cyc}\frac{2S}{a(b+c-a)}\geq\sqrt{\frac{3(8abc+(a+b-c)(a+c-b)(b+c-a))}{4abc}}$$ or
$$\sum_{cyc}\frac{2\sqrt{(x+y+z)xyz}}{2x(y+z)}\geq\sqrt{\frac{3(8(x+y)(x+z)(y+z)+8xyz)}{4(x+y)(x+z)(y+z)}}$$ or
$$\sum_{cyc}yz(x+y)(x+z)\geq\sqrt{6(xy+xz+yz)xyz(x+y)(x+z)(y+z)}.$$
Now, let $xy=r$, $xz=q$ and $yz=p$.
Thus, we need to prove that
$$\sum_{cyc}(p+q)(p+r)\geq\sqrt{6(p+q+r)(p+q)(p+r)(q+r)}$$ or
$$\sum_{sum}(p^4-p^2q^2)\geq0$$ or
$$\sum_{cyc}(p^2-q^2)^2\geq0.$$
Done!
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