<s>[INST] How prove this inequality for $a,b,c,d$ are real numbers let $a,b,c,d$ are real numbers,show that
$$2\sqrt{a^2+c^2}+\sqrt{a^2+c^2+3(b^2+d^2)-2\sqrt{3}(ab+cd)}+\sqrt{a^2+c^2+3(b^2+d^2)+2\sqrt{3}(ab+cd)}\ge6\sqrt{|ad-bc|}$$
This problem  is creat by  China's famous mathematician hua luogeng,http://en.wikipedia.org/wiki/Hua_Luogeng 
when he is child,and  Now this problem has some ugly methods,I think this inequality has nice methods,Thank you
 [/INST] HINT: 
$$2\sqrt{a^2+c^2}+\sqrt{a^2+c^2+3(b^2+d^2)-2\sqrt{3}(ab+cd)}+\sqrt{a^2+c^2+3(b^2+d^2)+2\sqrt{3}(ab+cd)}$$ $$=2\sqrt{a^2+c^2}+\sqrt{(a-\sqrt{3}b)^2+(c-\sqrt{3}d)^2}+\sqrt{(a+\sqrt{3}b)^2+(c+\sqrt{3}d)^2}$$ $$\geq 2\sqrt{2|ac|}+\sqrt{2|(a-\sqrt{3}b)(c-\sqrt{3}d)|}+\sqrt{2|(a+\sqrt{3}b)(c+\sqrt{3}d)|}$$
Now,
$$\sqrt{2|(a-\sqrt{3}b)(c-\sqrt{3}d)|}+\sqrt{2|(a+\sqrt{3}b)(c+\sqrt{3}d)|}\geq\sqrt{2(|(a-\sqrt{3}b)(c-\sqrt{3}d)|+|(a+\sqrt{3}b)(c+\sqrt{3}d)|)}$$ 
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