Variation of the sum of distances












1














Let $l$ be a line and $A$ and $B$ two points on the same side of $l$. To find the point $P$ for which $AP+PB$ is minimum we take the intersection of $l$ and the line joining $B$ and the symmetric $A'$ of $A$ with respect to $l$. For any point $M$ of $l$ other than $P$ we have $AM+MB=A'M+MB>A'B=AP+PB$ so $P$ is the desired point.



My question is $color{red}{text{how to prove that $AM+MB$ increases with $PM$?}}$.



My attempt: If $M'$ is another point of $l$ such that $PM'>PM$ then $AM<AM'+MM'$ and $BM<BM'+MM'$. I want to prove that $AM+MB<AM'+M'B$ using only the triangle inequality.






geometry euclidean-geometry triangle reflection







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    1














    Let $l$ be a line and $A$ and $B$ two points on the same side of $l$. To find the point $P$ for which $AP+PB$ is minimum we take the intersection of $l$ and the line joining $B$ and the symmetric $A'$ of $A$ with respect to $l$. For any point $M$ of $l$ other than $P$ we have $AM+MB=A'M+MB>A'B=AP+PB$ so $P$ is the desired point.



    My question is $color{red}{text{how to prove that $AM+MB$ increases with $PM$?}}$.



    My attempt: If $M'$ is another point of $l$ such that $PM'>PM$ then $AM<AM'+MM'$ and $BM<BM'+MM'$. I want to prove that $AM+MB<AM'+M'B$ using only the triangle inequality.






    geometry euclidean-geometry triangle reflection







    share|cite|improve this question



























      1












      1








      1







      Let $l$ be a line and $A$ and $B$ two points on the same side of $l$. To find the point $P$ for which $AP+PB$ is minimum we take the intersection of $l$ and the line joining $B$ and the symmetric $A'$ of $A$ with respect to $l$. For any point $M$ of $l$ other than $P$ we have $AM+MB=A'M+MB>A'B=AP+PB$ so $P$ is the desired point.



      My question is $color{red}{text{how to prove that $AM+MB$ increases with $PM$?}}$.



      My attempt: If $M'$ is another point of $l$ such that $PM'>PM$ then $AM<AM'+MM'$ and $BM<BM'+MM'$. I want to prove that $AM+MB<AM'+M'B$ using only the triangle inequality.






      geometry euclidean-geometry triangle reflection







      share|cite|improve this question















      Let $l$ be a line and $A$ and $B$ two points on the same side of $l$. To find the point $P$ for which $AP+PB$ is minimum we take the intersection of $l$ and the line joining $B$ and the symmetric $A'$ of $A$ with respect to $l$. For any point $M$ of $l$ other than $P$ we have $AM+MB=A'M+MB>A'B=AP+PB$ so $P$ is the desired point.



      My question is $color{red}{text{how to prove that $AM+MB$ increases with $PM$?}}$.



      My attempt: If $M'$ is another point of $l$ such that $PM'>PM$ then $AM<AM'+MM'$ and $BM<BM'+MM'$. I want to prove that $AM+MB<AM'+M'B$ using only the triangle inequality.






      geometry euclidean-geometry triangle reflection




      geometry euclidean-geometry triangle reflection






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 6 at 14:51







      Trump

















      asked Dec 30 '18 at 17:01









      TrumpTrump

      42




      42






















          2 Answers
          2






          active

          oldest

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          0














          Sometimes a figure is worth a thousand words:



          enter image description here



          This distance from $A$ to $B$ via the line is the same as the distance from $A$ to $B'$... the shortest of which is a straight line. Using the elementary fact from Euclidean geometry that the shortest distance between two points is a straight line we see:
          $color{red}{text{all other such paths (see green) must be longer.}}$ Proof completed.






          share|cite|improve this answer



















          • 1




            Using your image Let $M'$ be another point on the black line. Draw the path $AM'B'$ in blue. How to prove that if $PM'>PM$ then the blue path is longer then the green path?
            – Trump
            Dec 30 '18 at 17:38



















          0














          Your inequality is false: see diagram below for a counterexample.



          enter image description here






          share|cite|improve this answer























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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            0














            Sometimes a figure is worth a thousand words:



            enter image description here



            This distance from $A$ to $B$ via the line is the same as the distance from $A$ to $B'$... the shortest of which is a straight line. Using the elementary fact from Euclidean geometry that the shortest distance between two points is a straight line we see:
            $color{red}{text{all other such paths (see green) must be longer.}}$ Proof completed.






            share|cite|improve this answer



















            • 1




              Using your image Let $M'$ be another point on the black line. Draw the path $AM'B'$ in blue. How to prove that if $PM'>PM$ then the blue path is longer then the green path?
              – Trump
              Dec 30 '18 at 17:38
















            0














            Sometimes a figure is worth a thousand words:



            enter image description here



            This distance from $A$ to $B$ via the line is the same as the distance from $A$ to $B'$... the shortest of which is a straight line. Using the elementary fact from Euclidean geometry that the shortest distance between two points is a straight line we see:
            $color{red}{text{all other such paths (see green) must be longer.}}$ Proof completed.






            share|cite|improve this answer



















            • 1




              Using your image Let $M'$ be another point on the black line. Draw the path $AM'B'$ in blue. How to prove that if $PM'>PM$ then the blue path is longer then the green path?
              – Trump
              Dec 30 '18 at 17:38














            0












            0








            0






            Sometimes a figure is worth a thousand words:



            enter image description here



            This distance from $A$ to $B$ via the line is the same as the distance from $A$ to $B'$... the shortest of which is a straight line. Using the elementary fact from Euclidean geometry that the shortest distance between two points is a straight line we see:
            $color{red}{text{all other such paths (see green) must be longer.}}$ Proof completed.






            share|cite|improve this answer














            Sometimes a figure is worth a thousand words:



            enter image description here



            This distance from $A$ to $B$ via the line is the same as the distance from $A$ to $B'$... the shortest of which is a straight line. Using the elementary fact from Euclidean geometry that the shortest distance between two points is a straight line we see:
            $color{red}{text{all other such paths (see green) must be longer.}}$ Proof completed.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Dec 30 '18 at 17:36

























            answered Dec 30 '18 at 17:18









            David G. StorkDavid G. Stork

            10.1k21332




            10.1k21332








            • 1




              Using your image Let $M'$ be another point on the black line. Draw the path $AM'B'$ in blue. How to prove that if $PM'>PM$ then the blue path is longer then the green path?
              – Trump
              Dec 30 '18 at 17:38














            • 1




              Using your image Let $M'$ be another point on the black line. Draw the path $AM'B'$ in blue. How to prove that if $PM'>PM$ then the blue path is longer then the green path?
              – Trump
              Dec 30 '18 at 17:38








            1




            1




            Using your image Let $M'$ be another point on the black line. Draw the path $AM'B'$ in blue. How to prove that if $PM'>PM$ then the blue path is longer then the green path?
            – Trump
            Dec 30 '18 at 17:38




            Using your image Let $M'$ be another point on the black line. Draw the path $AM'B'$ in blue. How to prove that if $PM'>PM$ then the blue path is longer then the green path?
            – Trump
            Dec 30 '18 at 17:38











            0














            Your inequality is false: see diagram below for a counterexample.



            enter image description here






            share|cite|improve this answer




























              0














              Your inequality is false: see diagram below for a counterexample.



              enter image description here






              share|cite|improve this answer


























                0












                0








                0






                Your inequality is false: see diagram below for a counterexample.



                enter image description here






                share|cite|improve this answer














                Your inequality is false: see diagram below for a counterexample.



                enter image description here







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 2 days ago

























                answered Jan 6 at 17:43









                AretinoAretino

                22.8k21443




                22.8k21443






























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