Question

ABCD is quadrilateral. Is AB + BC + CD + DA >AC + BD?

Answer 1

In a triangle, the sum of the lengths of either two sides is always greater than the third side.
Considering \(ΔABC\),
\(AB + BC > CA\) (i)
In \(ΔBCD\),
\(BC + CD > DB\) (ii)
In \(ΔCDA\),
\(CD + DA > AC\) (iii)
In \(ΔDAB\),
\(DA + AB > DB\) (iv)
Adding equations (i), (ii), (iii), and (iv), we obtain
\(AB + BC + BC + CD + CD + DA + DA + AB > AC + BD + AC + BD\)
\(\Rightarrow 2AB + 2BC + 2CD +2DA > 2AC + 2BD\)
\(\Rightarrow2(AB + BC + CD + DA) > 2(AC + BD)\)
\(\Rightarrow(AB + BC + CD + DA) > (AC + BD)\) 

Yes, the given expression is true.