Prove that triangle midline and median split themselves into halves












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How can I prove that median and midline in a triangle split themselves into halves?



Thanks a lot in advance!










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  • $begingroup$
    Median I know, but what is "midline"?
    $endgroup$
    – DonAntonio
    Jan 20 at 21:26










  • $begingroup$
    @DonAntonio it connects centers of sides in a triangle
    $endgroup$
    – Xxx Ddd
    Jan 20 at 21:30










  • $begingroup$
    Ok, thanks. Do you know Thales theorems?
    $endgroup$
    – DonAntonio
    Jan 20 at 22:01










  • $begingroup$
    @DonAntonio yep, I know what that is
    $endgroup$
    – Xxx Ddd
    Jan 20 at 22:03
















0












$begingroup$


How can I prove that median and midline in a triangle split themselves into halves?



Thanks a lot in advance!










share|cite|improve this question









$endgroup$












  • $begingroup$
    Median I know, but what is "midline"?
    $endgroup$
    – DonAntonio
    Jan 20 at 21:26










  • $begingroup$
    @DonAntonio it connects centers of sides in a triangle
    $endgroup$
    – Xxx Ddd
    Jan 20 at 21:30










  • $begingroup$
    Ok, thanks. Do you know Thales theorems?
    $endgroup$
    – DonAntonio
    Jan 20 at 22:01










  • $begingroup$
    @DonAntonio yep, I know what that is
    $endgroup$
    – Xxx Ddd
    Jan 20 at 22:03














0












0








0





$begingroup$


How can I prove that median and midline in a triangle split themselves into halves?



Thanks a lot in advance!










share|cite|improve this question









$endgroup$




How can I prove that median and midline in a triangle split themselves into halves?



Thanks a lot in advance!







geometry triangle






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 20 at 21:17









Xxx DddXxx Ddd

354




354












  • $begingroup$
    Median I know, but what is "midline"?
    $endgroup$
    – DonAntonio
    Jan 20 at 21:26










  • $begingroup$
    @DonAntonio it connects centers of sides in a triangle
    $endgroup$
    – Xxx Ddd
    Jan 20 at 21:30










  • $begingroup$
    Ok, thanks. Do you know Thales theorems?
    $endgroup$
    – DonAntonio
    Jan 20 at 22:01










  • $begingroup$
    @DonAntonio yep, I know what that is
    $endgroup$
    – Xxx Ddd
    Jan 20 at 22:03


















  • $begingroup$
    Median I know, but what is "midline"?
    $endgroup$
    – DonAntonio
    Jan 20 at 21:26










  • $begingroup$
    @DonAntonio it connects centers of sides in a triangle
    $endgroup$
    – Xxx Ddd
    Jan 20 at 21:30










  • $begingroup$
    Ok, thanks. Do you know Thales theorems?
    $endgroup$
    – DonAntonio
    Jan 20 at 22:01










  • $begingroup$
    @DonAntonio yep, I know what that is
    $endgroup$
    – Xxx Ddd
    Jan 20 at 22:03
















$begingroup$
Median I know, but what is "midline"?
$endgroup$
– DonAntonio
Jan 20 at 21:26




$begingroup$
Median I know, but what is "midline"?
$endgroup$
– DonAntonio
Jan 20 at 21:26












$begingroup$
@DonAntonio it connects centers of sides in a triangle
$endgroup$
– Xxx Ddd
Jan 20 at 21:30




$begingroup$
@DonAntonio it connects centers of sides in a triangle
$endgroup$
– Xxx Ddd
Jan 20 at 21:30












$begingroup$
Ok, thanks. Do you know Thales theorems?
$endgroup$
– DonAntonio
Jan 20 at 22:01




$begingroup$
Ok, thanks. Do you know Thales theorems?
$endgroup$
– DonAntonio
Jan 20 at 22:01












$begingroup$
@DonAntonio yep, I know what that is
$endgroup$
– Xxx Ddd
Jan 20 at 22:03




$begingroup$
@DonAntonio yep, I know what that is
$endgroup$
– Xxx Ddd
Jan 20 at 22:03










2 Answers
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$begingroup$

HINT.




  • In a triangle a midline is parallel to the third side and is a half of that.


  • In a triangle a line through the midpoint of a side and parallel to another side, cuts the third side at its midpoint.







share|cite|improve this answer









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    0












    $begingroup$

    Let $;Delta ABC;$ be the triangle, $;EF;$ the midline joining the middle points of $;AB,,AC;$ and $;AM;$ the median to $;BC;$, and let $;K;$ be the point of intersection of $;EF;$ and $;AM;$ .



    Since $;EF||BC;$, Thales (the intercepts theorems for parallels) theorem tells us, applied twice, that



    $$text{On};;Delta ABM;:;frac{AE}{EB}=frac{AK}{KM}=color{red}{frac{EK}{BM}};,;;;;;;;text{On};Delta AMC;:frac{AK}{KM}=frac{AF}{FC}=color{red}{frac{KF}{MC}} $$



    Now just equate both red parts (which are equal!) and remember that $;BM=MC;$ ...






    share|cite|improve this answer









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      2 Answers
      2






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      2 Answers
      2






      active

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      active

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      0












      $begingroup$

      HINT.




      • In a triangle a midline is parallel to the third side and is a half of that.


      • In a triangle a line through the midpoint of a side and parallel to another side, cuts the third side at its midpoint.







      share|cite|improve this answer









      $endgroup$


















        0












        $begingroup$

        HINT.




        • In a triangle a midline is parallel to the third side and is a half of that.


        • In a triangle a line through the midpoint of a side and parallel to another side, cuts the third side at its midpoint.







        share|cite|improve this answer









        $endgroup$
















          0












          0








          0





          $begingroup$

          HINT.




          • In a triangle a midline is parallel to the third side and is a half of that.


          • In a triangle a line through the midpoint of a side and parallel to another side, cuts the third side at its midpoint.







          share|cite|improve this answer









          $endgroup$



          HINT.




          • In a triangle a midline is parallel to the third side and is a half of that.


          • In a triangle a line through the midpoint of a side and parallel to another side, cuts the third side at its midpoint.








          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 20 at 21:59









          AretinoAretino

          24.8k21444




          24.8k21444























              0












              $begingroup$

              Let $;Delta ABC;$ be the triangle, $;EF;$ the midline joining the middle points of $;AB,,AC;$ and $;AM;$ the median to $;BC;$, and let $;K;$ be the point of intersection of $;EF;$ and $;AM;$ .



              Since $;EF||BC;$, Thales (the intercepts theorems for parallels) theorem tells us, applied twice, that



              $$text{On};;Delta ABM;:;frac{AE}{EB}=frac{AK}{KM}=color{red}{frac{EK}{BM}};,;;;;;;;text{On};Delta AMC;:frac{AK}{KM}=frac{AF}{FC}=color{red}{frac{KF}{MC}} $$



              Now just equate both red parts (which are equal!) and remember that $;BM=MC;$ ...






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                Let $;Delta ABC;$ be the triangle, $;EF;$ the midline joining the middle points of $;AB,,AC;$ and $;AM;$ the median to $;BC;$, and let $;K;$ be the point of intersection of $;EF;$ and $;AM;$ .



                Since $;EF||BC;$, Thales (the intercepts theorems for parallels) theorem tells us, applied twice, that



                $$text{On};;Delta ABM;:;frac{AE}{EB}=frac{AK}{KM}=color{red}{frac{EK}{BM}};,;;;;;;;text{On};Delta AMC;:frac{AK}{KM}=frac{AF}{FC}=color{red}{frac{KF}{MC}} $$



                Now just equate both red parts (which are equal!) and remember that $;BM=MC;$ ...






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  Let $;Delta ABC;$ be the triangle, $;EF;$ the midline joining the middle points of $;AB,,AC;$ and $;AM;$ the median to $;BC;$, and let $;K;$ be the point of intersection of $;EF;$ and $;AM;$ .



                  Since $;EF||BC;$, Thales (the intercepts theorems for parallels) theorem tells us, applied twice, that



                  $$text{On};;Delta ABM;:;frac{AE}{EB}=frac{AK}{KM}=color{red}{frac{EK}{BM}};,;;;;;;;text{On};Delta AMC;:frac{AK}{KM}=frac{AF}{FC}=color{red}{frac{KF}{MC}} $$



                  Now just equate both red parts (which are equal!) and remember that $;BM=MC;$ ...






                  share|cite|improve this answer









                  $endgroup$



                  Let $;Delta ABC;$ be the triangle, $;EF;$ the midline joining the middle points of $;AB,,AC;$ and $;AM;$ the median to $;BC;$, and let $;K;$ be the point of intersection of $;EF;$ and $;AM;$ .



                  Since $;EF||BC;$, Thales (the intercepts theorems for parallels) theorem tells us, applied twice, that



                  $$text{On};;Delta ABM;:;frac{AE}{EB}=frac{AK}{KM}=color{red}{frac{EK}{BM}};,;;;;;;;text{On};Delta AMC;:frac{AK}{KM}=frac{AF}{FC}=color{red}{frac{KF}{MC}} $$



                  Now just equate both red parts (which are equal!) and remember that $;BM=MC;$ ...







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 20 at 22:12









                  DonAntonioDonAntonio

                  179k1494232




                  179k1494232






























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