Ash and his brother Jerry propose to go for a local Football Match. Ash sets out to the ground straight, while Jerry wants to pick up his friend Monika to the match. The topographic positions of Ash's House, Monika's house and the Foot Ball ground are shown below.
If the question is asked who is expected to reach the ground first, Ash or his brother Jerry, then we are intuitive enough to say that Ash will reach the ground first. We expect that the the sum of the distances Jerry needs to cover is longer than the straight length Ash is moving on. The triangle inequality theorem states exactly this fact.
It states that, the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
|| AB + BC > AC
BC + AC > AB
AC + AB > BC
Let us now prove this theorem using a construction.