question_answer 1)

\[ABCD\] is a quadrilateral. Is \[AB+BC+CD+DA<2(AC+BD)\]?

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), \[\text{2(OA}+\text{OB}+\text{OC}+\text{OD)}>\text{AB}+\text{BC}+\text{CD}+\text{DA}\] \[\Rightarrow \]               \[\text{AB}+\text{BC}+\text{CD}+\text{DA}<\text{2}\] \[\text{(OA}+\text{OB}+\text{OC}+\text{OD)}\] \[\Rightarrow \]               \[~\text{AB}+\text{BC}+\text{CD}+\text{DA}<\text{2}\] \[\text{(OA}+\text{OC}+\text{OB}+\text{OD)}\] \[\Rightarrow \]               \[\text{AB}+\text{BC}+\text{CD}+\text{DA}<\text{2(AC}+\text{BD)}\].