a line segment has at most one midpoint

(this proof is not correct yet)


Let [p,q] be a closed line segment and suppose m and m are midpointsMathworldPlanetmathPlanetmathPlanetmath. 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:q. Suppose mm. Without loss of generality we can assume p:m:m and m:m:q. But then [p,m]>[p,m][m,q]>[m,q] so that [p,m]≇[m,q], a contradictionMathworldPlanetmathPlanetmath. Hence m=m. ∎

Title a line segment has at most one midpoint
Canonical name ALineSegmentHasAtMostOneMidpoint
Date of creation 2013-03-22 17:17:37
Last modified on 2013-03-22 17:17:37
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 9
Author Mathprof (13753)
Entry type Theorem
Classification msc 51G05