What does the shortest distance between them have to do with the triangle Inequality Conjecture?

The shortest distance between any two points, A and B, in a plane is the straight line joining them.

Suppose, that the distance A to C and then C to B is shorter where C is any point not on AB. That would imply that, in triangle ABC, the sum of the lengths of two sides (AC and CB) is shorter tan the third side (AB). That contradicts the inequality conjecture.