I'm currently reading Pugh's Analysis. He makes the statement that the line between two points x and y is the set of linear combinations s x + t y where s + t = 1. I'm satisfied that this is true, as the line between two points x and y has the equation y − x 2 = x 2 − y 2 x 1 − y 1 ( x − x 1 ) and the points of form ( s x 1 + t y 1 , s x 2 + t y 2 ) are solutions of the equation. I still don't have any reasonable geometric intuition for why this is true though.

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I&#039;m currently reading Pugh&#039;s Analysis. He makes the statement that the line between two points x and y is the set of linear combinations   s  x  +  t  y where   s  +  t  =  1. I&#039;m satisfied that this is true, as the line between two points   x and   y has the equation   y  −      x    2    =                    x        2            −              y        2                            x        1            −              y        1              (  x  −      x    1    ) and the points of form   (  s      x    1    +  t      y    1    ,  s      x    2    +  t      y    2    ) are solutions of the equation. I still don&#039;t have any reasonable geometric intuition for why this is true though.

Ryan Newman · Accepted Answer

It’s easiest to see what’s going on when   0  ≤  s  ,  t  ≤  1. In that case   s  x  +  t  y is a weighted mean of   x and   y. When   s  =  t  =      1    2  , for instance, it’s the ordinary arithmetic mean, and the point       1    2    x  +      1    2    y is the midpoint of             x      y        ¯  . When   s  =      1    3   and   t  =      2    3  ,   y is given twice the weight of   x, so the point is ‘twice as close’ to   y as it is to   x: in more understandable terms, it’s half as far from   y as it is from   x, so that it’s       2    3   of the way from   x to   y. In general,   s  x  +  t  y is   t fraction of the way from   x to   y when   0  ≤  s  ,  t  ≤  1.Once   s and   t get outside the range   [  0  ,  1  ], it’s harder to get an intuitive picture, but you can still regard   s  x  +  y  t is a kind of weighted mean in which one of the weights is negative. Thus,   −  x  +  2  y  =  y  +  (  y  −  x  ) is the point that is   y  −  x units past   y in the direction away from   x, and in general   s  x  +  t  y is the point that is   y  −  x units past   (  t  −  1  )  y in the direction away from x when   t  &amp;gt;  1. (And of course everything just turns around when   s  &amp;gt;  1.)

I'm currently reading Pugh's Analysis. He makes the statement that the line between two points x and

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