question_answer 1)

AM is a median of a triangle ABC.                             Is \[AB+BC+CA>2AM?\] (Consider the sides of triangles \[\Delta ABM\] and \[\Delta AMC\].)                

Answer:

                In \[\Delta ABM\],                 \[AB+BM>AM\]                               \[\left| \begin{align}   & \text{Sum of the length of any two side of a triangle } \\  & \text{is greater than the lenth of the third side} \\ \end{align} \right.\]        In \[\Delta ACM\], \[CA+CM>AM\]                                               ? (2) \[\left| \begin{align}   & \text{Sum of the lenths of any two sides of a triangle} \\  & \text{is greater than the lenth of the third side} \\ \end{align} \right.\] Sum (1) and (2), \[(AB+BM)+(CA+CM)>AM+AM\] \[\Rightarrow \]               \[AB+(BM+CM)+CA>2AM\] \[\Rightarrow \]               \[AB+BC+CA>2AM\]