Prove that $CE=AB$

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Suppose $ABC$ is an acute-angled triangle with $AB<AC$.Let $M$ be the midpoint of $BC$. Suppose $P$ is a point on side $AB$ such that, if $PC$ intersects the median $AM$ at E, then $AP=PE$.Prove that $AB=CE$.




I don't know how to start. Please give me an idea. Getting no fruitful thoughts ,I started using barycentric Coordinates.But the calculations seemed very tough and I failed to proceed. Please give me any idea to start




geometry euclidean-geometry triangle




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    Suppose $ABC$ is an acute-angled triangle with $AB<AC$.Let $M$ be the midpoint of $BC$. Suppose $P$ is a point on side $AB$ such that, if $PC$ intersects the median $AM$ at E, then $AP=PE$.Prove that $AB=CE$.




    I don't know how to start. Please give me an idea. Getting no fruitful thoughts ,I started using barycentric Coordinates.But the calculations seemed very tough and I failed to proceed. Please give me any idea to start




    geometry euclidean-geometry triangle




    share|cite|improve this question





















      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite












      Suppose $ABC$ is an acute-angled triangle with $AB<AC$.Let $M$ be the midpoint of $BC$. Suppose $P$ is a point on side $AB$ such that, if $PC$ intersects the median $AM$ at E, then $AP=PE$.Prove that $AB=CE$.




      I don't know how to start. Please give me an idea. Getting no fruitful thoughts ,I started using barycentric Coordinates.But the calculations seemed very tough and I failed to proceed. Please give me any idea to start




      geometry euclidean-geometry triangle




      share|cite|improve this question












      Suppose $ABC$ is an acute-angled triangle with $AB<AC$.Let $M$ be the midpoint of $BC$. Suppose $P$ is a point on side $AB$ such that, if $PC$ intersects the median $AM$ at E, then $AP=PE$.Prove that $AB=CE$.




      I don't know how to start. Please give me an idea. Getting no fruitful thoughts ,I started using barycentric Coordinates.But the calculations seemed very tough and I failed to proceed. Please give me any idea to start





      geometry euclidean-geometry triangle





      share|cite|improve this question










      share|cite|improve this question




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      asked Jul 23 at 17:23









      Sufaid Saleel

      1,666625




      1,666625




















          2 Answers
          2






          active

          oldest

          votes

















          up vote
          4
          down vote



          accepted










          Hint:   write Menelaus' theorem for triangle $,triangle PBC,$ and transversal $,AM,$.






          share|cite|improve this answer




























            up vote
            2
            down vote













            Construct the //gm BECF.



            enter image description here



            From (1) AEM is a straight line; (2) E is a vertex of that //gm; (3) M is the midpoint of one of its diagonals; and (4) F is a vertex of that //gm and FM is a straight line, we can say that AEMF is a straight line.



            The required result follows because all the green marked angles are equal.






            share|cite|improve this answer























            • What does //gm mean?
              – greedoid
              Jul 23 at 19:02











            • @Angle I guess it stands for Parallelogram?
              – Mythomorphic
              Jul 23 at 19:13










            • Nice solution +1
              – greedoid
              Jul 23 at 19:25










            • @Mythomorphic That is right.
              – Mick
              Jul 24 at 2:45










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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            4
            down vote



            accepted










            Hint:   write Menelaus' theorem for triangle $,triangle PBC,$ and transversal $,AM,$.






            share|cite|improve this answer

























              up vote
              4
              down vote



              accepted










              Hint:   write Menelaus' theorem for triangle $,triangle PBC,$ and transversal $,AM,$.






              share|cite|improve this answer























                up vote
                4
                down vote



                accepted







                up vote
                4
                down vote



                accepted






                Hint:   write Menelaus' theorem for triangle $,triangle PBC,$ and transversal $,AM,$.






                share|cite|improve this answer













                Hint:   write Menelaus' theorem for triangle $,triangle PBC,$ and transversal $,AM,$.







                share|cite|improve this answer













                share|cite|improve this answer



                share|cite|improve this answer











                answered Jul 23 at 17:31









                dxiv

                54k64796




                54k64796




















                    up vote
                    2
                    down vote













                    Construct the //gm BECF.



                    enter image description here



                    From (1) AEM is a straight line; (2) E is a vertex of that //gm; (3) M is the midpoint of one of its diagonals; and (4) F is a vertex of that //gm and FM is a straight line, we can say that AEMF is a straight line.



                    The required result follows because all the green marked angles are equal.






                    share|cite|improve this answer























                    • What does //gm mean?
                      – greedoid
                      Jul 23 at 19:02











                    • @Angle I guess it stands for Parallelogram?
                      – Mythomorphic
                      Jul 23 at 19:13










                    • Nice solution +1
                      – greedoid
                      Jul 23 at 19:25










                    • @Mythomorphic That is right.
                      – Mick
                      Jul 24 at 2:45














                    up vote
                    2
                    down vote













                    Construct the //gm BECF.



                    enter image description here



                    From (1) AEM is a straight line; (2) E is a vertex of that //gm; (3) M is the midpoint of one of its diagonals; and (4) F is a vertex of that //gm and FM is a straight line, we can say that AEMF is a straight line.



                    The required result follows because all the green marked angles are equal.






                    share|cite|improve this answer























                    • What does //gm mean?
                      – greedoid
                      Jul 23 at 19:02











                    • @Angle I guess it stands for Parallelogram?
                      – Mythomorphic
                      Jul 23 at 19:13










                    • Nice solution +1
                      – greedoid
                      Jul 23 at 19:25










                    • @Mythomorphic That is right.
                      – Mick
                      Jul 24 at 2:45












                    up vote
                    2
                    down vote










                    up vote
                    2
                    down vote









                    Construct the //gm BECF.



                    enter image description here



                    From (1) AEM is a straight line; (2) E is a vertex of that //gm; (3) M is the midpoint of one of its diagonals; and (4) F is a vertex of that //gm and FM is a straight line, we can say that AEMF is a straight line.



                    The required result follows because all the green marked angles are equal.






                    share|cite|improve this answer















                    Construct the //gm BECF.



                    enter image description here



                    From (1) AEM is a straight line; (2) E is a vertex of that //gm; (3) M is the midpoint of one of its diagonals; and (4) F is a vertex of that //gm and FM is a straight line, we can say that AEMF is a straight line.



                    The required result follows because all the green marked angles are equal.







                    share|cite|improve this answer















                    share|cite|improve this answer



                    share|cite|improve this answer








                    edited Jul 23 at 18:53


























                    answered Jul 23 at 18:45









                    Mick

                    11.5k21540




                    11.5k21540











                    • What does //gm mean?
                      – greedoid
                      Jul 23 at 19:02











                    • @Angle I guess it stands for Parallelogram?
                      – Mythomorphic
                      Jul 23 at 19:13










                    • Nice solution +1
                      – greedoid
                      Jul 23 at 19:25










                    • @Mythomorphic That is right.
                      – Mick
                      Jul 24 at 2:45
















                    • What does //gm mean?
                      – greedoid
                      Jul 23 at 19:02











                    • @Angle I guess it stands for Parallelogram?
                      – Mythomorphic
                      Jul 23 at 19:13










                    • Nice solution +1
                      – greedoid
                      Jul 23 at 19:25










                    • @Mythomorphic That is right.
                      – Mick
                      Jul 24 at 2:45















                    What does //gm mean?
                    – greedoid
                    Jul 23 at 19:02





                    What does //gm mean?
                    – greedoid
                    Jul 23 at 19:02













                    @Angle I guess it stands for Parallelogram?
                    – Mythomorphic
                    Jul 23 at 19:13




                    @Angle I guess it stands for Parallelogram?
                    – Mythomorphic
                    Jul 23 at 19:13












                    Nice solution +1
                    – greedoid
                    Jul 23 at 19:25




                    Nice solution +1
                    – greedoid
                    Jul 23 at 19:25












                    @Mythomorphic That is right.
                    – Mick
                    Jul 24 at 2:45




                    @Mythomorphic That is right.
                    – Mick
                    Jul 24 at 2:45












                     

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