0

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.

lioness99a
  • 4,943
user2899944
  • 275
  • 1
    Sum of length of two sides of a triangle must be greater than third side...Not equal.. In your case , it is a degenerate triangle...Looks like a line segment – LM2357 Mar 17 '17 at 09:57

2 Answers2

2

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 .

Emilio Novati
  • 62,675
0

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.

Jaap Scherphuis
  • 13,660