# a line segment has at most one midpoint

(this proof is not correct yet)

###### 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 $m=m^{\prime}$. ∎

Title a line segment has at most one midpoint ALineSegmentHasAtMostOneMidpoint 2013-03-22 17:17:37 2013-03-22 17:17:37 Mathprof (13753) Mathprof (13753) 9 Mathprof (13753) Theorem msc 51G05