0
$\begingroup$

The line segment $\overline{AB}$ is defined as $\{x \in U;X=A, or X=B, or A-X-B\}$ such that $U$ is the universal set of all points.

Now consider two distinct midpoints $C$ and $D$ that are elements of the segment $AB$ such that $C$ and $D$ are between $\overline{AB}$ By the definition of midpoint you have that $\overline{AC}=\overline{CB}$ and that $\overline{AD}=\overline{DB}$. Also $\overline{AC}+\overline{BC}=\overline{AB}$ and $\overline{AD}+\overline{DB}= \overline{AB}$

Then $\overline{AC}+\overline{BC}=\overline{AD}+\overline{DB}$

Then $2\overline{AC}=2\overline{AD}$

Therefore $\overline{AC}=\overline{AD}$

So I want to thus conclude that $C=D$ which is a contradiction so therefore there exists only one midpoint for every line segment.

I need to show existence and uniqueness I don't believe I showed existence very well.

  • 0
    What does $A-X-B$ mean? – 2017-02-06
  • 0
    It means that $X$ is between both $A$ and $B$ – 2017-02-06
  • 0
    `I don't believe I showed existence very well` The midpoint can be constructed, see for example Euclid's [book I proposition X](http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI10.html). (Of course, this may depend on the axiomatic system you work within.) – 2017-02-07

0 Answers 0