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# a line segment has at most one midpoint

(this proof is not correct yet)

###### Theorem 1.

In an ordered geometry a line segment has at most one midpoint.

###### Proof.

Let $[p,q]$ be a closed line segment and suppose $m$ and $m^{{\prime}}$ are midpoints. If $m:p:q$ then $[m,p]<[m,q]$ so $m$ is not a midpoint. Similarly we cannot have $p:q:m$, so we have $p:m:q$. And also, $p:m^{{\prime}}:q$. Suppose $m\not=m^{{\prime}}$. Without loss of generality we can assume $p:m:m^{{\prime}}$ and $m:m^{{\prime}}:q$. But then $[p,m^{{\prime}}]>[p,m]\cong[m,q]>[m^{{\prime}},q]$ so that $[p,m^{{\prime}}]\not\cong[m^{{\prime}},q]$, a contradiction. Hence $m=m^{{\prime}}$. ∎

Type of Math Object:

Theorem

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## Mathematics Subject Classification

51G05*no label found*

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