$A B C D$ is a Quadrilateral.Is $A B + B C + C D + D A < 2 (A C + B D)$ ?

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Solution

By triangle property,
$\begin{matrix} O A + O B > A B O C + O B > B C O C + O D > C D O D + O A > A D \end{matrix}$
Adding all,
$\begin{matrix} 2 (O A + O B + O C + O D) > A B + B C + C D + D A 2 (O A + O C - --------- - + O B + O D - --------- -) 2 (A C + B D) > A B + B C + C D + D A A B + B C + C D + D A < 2 (A C + B D) \end{matrix}$

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