See image for naming of the sides: the medians are
ab, bc, and ac; and sides are a, b, and c
Pick one median, say, ab. The following are true:
ab < a
ab < b
->
ab + ab < a + b
ab < 0.5*(a+b) | add 0.5*c to the right hand side (does not change inequality)
ab < 0.5(a+b+c) = half perimeter
so ab < half perimeter
Same argument for bc, and ac holds.
