How to prove that 3 points are on the same line by the distance between the points

Question

If I am given 3 points, e.g. $A=(-2,-1)$, $B=(1,3)$, $C=(7,11)$ and I wish to prove that they are all on the same line using the distance between all pairs, how do I do that?

I know that $AB=5$, $AC=15$ and $BC=10$.

According to the triangle inequality, the sum of two lines can be equal to the third line and it will still be a triangle. What am I missing here? I don't wish to create the equation of a line and then try putting the third point in it to show that it works, I want to use distances only. Thank you.

Answer 1

Hint:

The triangle inequality says that if the three sides of a triangle have lengths $x\le y \le z$ than $x+y\ge z$ and the equality only applies if the three vertices are aligned and the triangle is degenerate .

Answer 2

Travelling from A to C directly (i.e. in a straight line) is usually shorter than if you had to take a detour to visit B. If going via B is not any longer than the direct trip, then B must actually be on the route from A to C, i.e. all three points lie on a straight line.