Answer:
In \[\Delta OAB,\] \[OA+OB>AB\] ? (1) \[\left| \begin{align} & \text{Sum of the lenth of any two sides of a triangle is} \\ & \text{greater than the length of the third side}\text{. } \\ \end{align} \right.\] In \[\Delta {\mathrm O}\Beta C,OB+OC>BC\] ? (2) \[\left| \begin{align} & \text{Sum of the lengths of any two sides of a triangle} \\ & \text{is greater than the length of the third side} \\ \end{align} \right.\] In \[\Delta OCA,OC+OA>CA\] ? (3) \[\left| \begin{align} & \text{Sum of the lengths of any two sides of a triangle is} \\ & \text{greater than the length of the third side} \\ \end{align} \right.\] In \[\Delta OAD,OA+OD>AD\] ? (4) \[\left| \begin{align} & \text{Sum of the lengths of any two sides of a triangle is} \\ & \text{greater than the length of the third side} \\ \end{align} \right.\] Adding (1), (2), (3) and (4), \[\text{2(OA}+\text{OB}+\text{OC}+\text{OD)}>\text{AB}+\text{BC}+\text{CD}+\text{DA}\] \[\Rightarrow \] \[\text{AB}+\text{BC}+\text{CD}+\text{DA}<\text{2}\] \[\text{(OA}+\text{OB}+\text{OC}+\text{OD)}\] \[\Rightarrow \] \[~\text{AB}+\text{BC}+\text{CD}+\text{DA}<\text{2}\] \[\text{(OA}+\text{OC}+\text{OB}+\text{OD)}\] \[\Rightarrow \] \[\text{AB}+\text{BC}+\text{CD}+\text{DA}<\text{2(AC}+\text{BD)}\].
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