VT=$\sqrt{(\frac{a}{\sqrt{2}}-b)^{2}+(\frac{a}{\sqrt{2}})^{2}}$+$\sqrt{(b-\frac{\sqrt{3}}{2}c)^{2}+(\frac{c}{2})^{2}}$Đặt $\overrightarrow{u}$($\frac{a}{\sqrt{2}}$-b;$\frac{a}{\sqrt{2}}$) ; $\overrightarrow{v}$(b-$\frac{\sqrt{3}}{2}$c;$\frac{c}{2}$)VT=$\left| {\overrightarrow{u}} \right|$+$\left| {\overrightarrow{v}} \right|$$\geq$$\left| {\overrightarrow{u}+\overrightarrow{v}} \right|$(1)Mà $\left| {\overrightarrow{u}+\overrightarrow{v}} \right|$=$\sqrt{(\frac{a}{\sqrt{2}}-\frac{\sqrt{3}}{2}c)^{2}+(\frac{a}{\sqrt{2}}+\frac{c}{2})^{2}}$=$\sqrt{\frac{a^{2}}{2}+\frac{3}{4}c^{2}+\frac{a^{2}}{2}+\frac{c^{2}}{4}-\frac{\sqrt{3}}{\sqrt{2}}ac+\frac{1}{\sqrt{2}}ac}$=$\sqrt{a^{2}-\sqrt{2-\sqrt{3}}ac+c^{2}}$(2)Từ(1)&(2)$\Rightarrow$đpcm
VT=
$\sqrt{(\frac{a}{\sqrt{2}}-b)^{2}+(\frac{a}{\sqrt{2}})^{2}}$
+
$\sqrt{(b-\frac{\sqrt{3}}{2}c)^{2}+(\frac{c}{2})^{2}}$Đặt $\overrightarrow{u}$($\frac{a}{\sqrt{2}}$-b;$\frac{a}{\sqrt{2}}$) ; $\overrightarrow{v}$(b-$\frac{\sqrt{3}}{2}$c;$\frac{c}{2}$)VT=$\left| {\overrightarrow{u}} \right|$+$\left| {\overrightarrow{v}} \right|$$\geq$$\left| {\overrightarrow{u}+\overrightarrow{v}} \right|$(1)Mà $\left| {\overrightarrow{u}+\overrightarrow{v}} \right|$=$\sqrt{(\frac{a}{\sqrt{2}}-\frac{\sqrt{3}}{2}c)^{2}+(\frac{a}{\sqrt{2}}+\frac{c}{2})^{2}}$=$\sqrt{\frac{a^{2}}{2}+\frac{3}{4}c^{2}+\frac{a^{2}}{2}+\frac{c^{2}}{4}-\frac{\sqrt{3}}{\sqrt{2}}ac+\frac{1}{\sqrt{2}}ac}$=$\sqrt{a^{2}-\sqrt{2-\sqrt{3}}ac+c^{2}}$(2)Từ(1)&(2)$\Rightarrow$đpcm