Question

AM is a median of a triangle ABC. Is AB + BC + CA $>$ 2AM?

Answer 1

AM is the median drawn from the vertex A to BC,

Applying the triangle inequality in the triangle ABM
AB + BM $>$ AM —---(i)

Applying the triangle inequality in the triangle AMC
AC + MC $>$ AM -----(ii)

Adding equations (i) and (ii) we get,
AB + BM + AC + MC $>$ AM + AM
AB + AC + (BM + MC) $>$ 2AM
AB + AC + BC $>$ 2AM