Proving the Triangle Inequality Theorem using Coordinates

[siliang]

Homework Statement

Prove the Triangle Inequality Theorum using the coordinate system.

Homework Equations

The corners of the triangles will be at (x1,y1), (x2, y2), (x3,y3)

The Attempt at a Solution

The proof that I know is proving that |x+y|<=|x|+|y|:

-|x|<x<|x|, and -|y|<y<|y|
then -|x|-|y|<x+y<|x|+|y|
absolute value proterty yields |x+y|<=|x|+|y|

I have absolutely no idea how to incorporate the coordinate system into my proof. I would assume the distance formula has something to do with it. Can someone please help? > . <

 

[Char. Limit]

Well, take the distance along the longest side. (I'm going to assume that x₁ to x₃ is the side that is longest) So, you know the formula for distance between those two points. Then prove that this distance is less than the sum of the lengths of the x₁-x₂ line and the x₂-x₃ line.

 

[SammyS]

How does the distance from (x₁, y₁) to (x₂, y₂)

PLUS

the distance from (x₂, y₂) to (x₃, y₃)

compare to

the distance from (x₁, y₁) to (x₃, y₃)

?

 

[siliang]

the triangle inequality states that the sum of any two sides of a triangle is larger than the third. Thanks for the help. But what if I wanted to prove is that for ANY given triangle or for any given 3 sides of a triangle, the triangle inequality applies?

 

[Char. Limit]

the triangle inequality states that the sum of any two sides of a triangle is larger than the third. Thanks for the help. But what if I wanted to prove is that for ANY given triangle or for any given 3 sides of a triangle, the triangle inequality applies?

Just do the same process for all three cases. Prove for each side that the length of that side is less than the length of the other two.