Adding up all median lengths of a triangle not exceeding perimeter?

Question

When adding up all lengths of the medians of a triangle, why does the result not exceed perimeter?

Answer 1

Let $D,E,F$ be the midpoints of $BC,CA,AB$ respectively

Hint: Apply the triangle inequality to triangles $ADE$, $BEF$ and $CFD$.

Use the fact that $|DE| = \frac{|AB|}{2}$ since they are midpoints.

Answer 2

Use the property that in a triangle ABC if AD , BE and CF are the three medians then 2AD ≤ AB + AC 2BE ≤ BC +BA 2CF ≤ CA +CB Add AD + BE + CF ≤ AB + BC +AC