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I am new to mathematics, so I apologize in advance if this question is trivial. I was trying to prove a property of an arbitrary three-point system in $\mathbb{R}^2$ regarding convexity. I tried it without looking at the solution, but to no avail. I looked at the solution and the author assumed WLOG $x$ to be the origin and the proof was really easy.

My question is why can we assume $x$ to be the origin. I think it has something to do with the function $f(a) = x - a$, where $x$ is one of the points in the three-point system, but I know there's more to it. I would like a deeper understanding of what's going on here. Anything is much appreciated.

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    $\begingroup$ Seems like it would help to know what property you were trying to prove. $\endgroup$
    – Zev Chonoles
    Jul 28, 2013 at 0:38
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    $\begingroup$ Presumably because whether a body is convex or not does not depend on where it is, so we may as well slide it over so one of its points is at the origin. Or something akin to this. I say "presumably" because I do not know what "a property" is supposed to refer to. $\endgroup$
    – anon
    Jul 28, 2013 at 0:38
  • $\begingroup$ By the way, please see here for a guide to writing math with MathJax, and see here for a guide to formatting posts with Markdown. $\endgroup$
    – Zev Chonoles
    Jul 28, 2013 at 0:38
  • $\begingroup$ You need $f(x)=x-a$, not $f(a)=x-a$. $\endgroup$
    – Michael Hardy
    Jul 28, 2013 at 0:43
  • $\begingroup$ This is a good question but it might help if we could see the example you're referring to. @anon's comment is probably very close to the reason though. $\endgroup$
    – Dan Rust
    Jul 28, 2013 at 0:44

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You’ve not really given us enough information to be able to answer the question with certainty. However, if the property of three-point systems that was to be proved is one that depends only on the relative positions of the points and not on their absolute positions in the Cartesian plane, then you can set the origin of your Cartesian coordinate system anywhere in the plane determined by the three points without affecting whether or not the points have the property. In particular, you can set the origin at one of the points, and doing so may make the calculations simpler. In some other context it might be more convenient to set the origin at the incentre, circumcentre, centroid, or orthocentre of the triangle determined by the three points.

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