July 27th, 2018, 07:55 AM  #1 
Senior Member Joined: Nov 2011 Posts: 250 Thanks: 3  the quick path
I have 3 point on a perimeter of circle. How I prove/show between every the two point of them there is always short way that the way between the other points? Last edited by shaharhada; July 27th, 2018 at 08:25 AM. 
July 27th, 2018, 08:02 AM  #2 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 
Can you please ask someone to help you with your English? Thank you. 
July 27th, 2018, 10:26 AM  #3 
Senior Member Joined: Nov 2011 Posts: 250 Thanks: 3 
I think the answer is the distance can be between infinitesimal (close to zero but not zero) to D (D = diameter) between 2 point on the perimeter of the circle. But (1) How I prove that when the distance is D, so the distance between every two points (that one is the point of the diameter) is smaller? (2) How I prove when 3 points are given the segments are follow the triangle inequality? Last edited by skipjack; July 27th, 2018 at 02:24 PM. 
July 27th, 2018, 02:25 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,973 Thanks: 2224 
Can you provide a diagram and state algebraically what you want to prove?

July 27th, 2018, 09:02 PM  #5 
Senior Member Joined: Nov 2011 Posts: 250 Thanks: 3 
What the blue segment (AB) is the maximum segment on the perimeter of the circle? What any other segment can't be bigger than AB. For example, the black segment (the AL segment)? Last edited by skipjack; July 27th, 2018 at 11:32 PM. 
July 27th, 2018, 11:51 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,973 Thanks: 2224  Segments.png Have you considered AL < AO + OL = AO + OB = AB? 
July 28th, 2018, 02:34 AM  #7 
Senior Member Joined: Nov 2011 Posts: 250 Thanks: 3 
Yes. It axiom or a theorem? If it is a theorem, how prove it? Last edited by skipjack; July 28th, 2018 at 03:01 AM. 
July 28th, 2018, 03:02 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 20,973 Thanks: 2224 
What exactly are you referring to as "it"?

July 28th, 2018, 03:12 AM  #9 
Senior Member Joined: Nov 2011 Posts: 250 Thanks: 3 
How I prove it by your picture? It take me a minute to look on the picture. I think you use Triangle inequality. But you prove it in another me as a I see. (1) Can it will be proven triangle quality? (2) Can you write it in tied up way the step of what you draw? 
July 28th, 2018, 03:33 AM  #10 
Global Moderator Joined: Dec 2006 Posts: 20,973 Thanks: 2224 
Applying the triangle inequality theorem to triangle AOL gives AL < AO + OL. If AB is a diameter of the circle and O is the midpoint of AB, OA, OB and OL are radii of the circle and have the same length. 

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