March 23rd, 2018, 11:06 AM  #1 
Banned Camp Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3  triangle inequality
My proof to the triangle inequality: (1) I draw two point A & B. (2) By the first axiom of Eculid: "between 2 points you can draw only one straight line'. (3) I draw a line between A & B. (4) => This line is the shortest distance between A & B. (Euclid axiom) (5) If i draw a broken line between A & B (Zigzag line), this line is longer becuase of (4). (6) So the line that pass A & B & C (C is the point that the vertix of the line). The triangle A & B & C  is inequality triagnle. 
March 23rd, 2018, 11:10 AM  #2 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,629 Thanks: 954 
All triangles are half a rectangle: isn't that enough?

March 23rd, 2018, 11:25 AM  #3 
Banned Camp Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 
I don't understand what you mean... Why? Is My proof not simple? You can explain what you mean... please!!!! 
March 23rd, 2018, 11:37 AM  #4 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,922 Thanks: 775 Math Focus: Wibbly wobbly timeywimey stuff.  
March 23rd, 2018, 12:25 PM  #5 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,629 Thanks: 954  
March 23rd, 2018, 12:42 PM  #6 
Banned Camp Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 
After I ask a colleague I have some notes: 1. the triangle ineqaulity in the metric space is axiom. 2. The Euclid space is example to metric space. There are another examples of other spaces. To prove that a space is Euclid space I need to prove the 3 axioms of the Euclid space that are true and especially triangle inequality. 3. My proof is not execptable proof. The axiom of between to points pass only one line don't tell nothing one their lenghts (even if is seem intuitive). The proof that the trangle ineqaulity is been in the Euclid space need to be a much more work. Actually, I need to use the inequality of CauchySwartz. So can one's show me the way to prove it by inequality of CauchySwartz? 

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