AM is a median of a triangle ABC. Is AB+BC+CA>2AM?
In triangle ABC, AM is a median.
In triangle ABM, AB+BM>AM… (1) [the sum of two sides of a triangle are greater than the third side].
In triangle ACM, AC+CM>AM… (2) [the sum of two sides of a triangle are greater than the third side].
Add (1) and (2) to get
AB+BM+AC+CM>AM+AM, or
AB+(BM+CM)+AC>2AM, or
AB+BC+CA>2AM. Proved