Question

Prove that any median in a triangle is less than the half of the triangle's perimeter.

Answer 1

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.