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In triangle ABC, the medians AD, BE, and CF concur at the centroid G.

 Prove that AD<(AB+AC)/2.

Guest Feb 21, 2017

Best Answer 

 #1
avatar+26366 
+10

Cut the triangle in two and rotate one part to make a new triangle as below (I've changed the notation somewhat):

 

Now we must have 2m < u + v   or   m < (u + v)/2  

 

or  AD < (AB + AC)/2

.

Alan  Feb 23, 2017
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4+0 Answers

 #1
avatar+26366 
+10
Best Answer

Cut the triangle in two and rotate one part to make a new triangle as below (I've changed the notation somewhat):

 

Now we must have 2m < u + v   or   m < (u + v)/2  

 

or  AD < (AB + AC)/2

.

Alan  Feb 23, 2017
 #2
avatar+79881 
0

Very nice, Alan.... genius in simplicity....!!!!!

 

 

 

cool cool cool

CPhill  Feb 23, 2017
 #3
avatar+91239 
0

Yes Alan thanks.  That is a really simple way to demonstrate that this relationship must be true and I am also impressed.

 

But

I am trying to work out how this could be proven in a formal proof......

 

I suppose you could write it exactly as you have done.

 

--------------------------

Consider the triangles ADB and ACD

 

AD bisects BC                       by definition of a median

therefore     BD=DC                                    

<BDA=180 - <ADC                  adjacent supplementary angles

 

Rotate triangle ADC 180 degrees about the point D. to from the new triangle AA'B   

AC has been rotated to the position A'B

i.e  AC=A'B     

 

AA' < A'B+AB                   One side of any triangle must be less than the sum of the other 2 sides.

AD+DA' < A'B+AB  

AD+AD < AC+AB

2AD < AC+AB

2AD < (AC+AB)/2

 

Would this pass as a formal proof?

Melody  Feb 24, 2017
 #4
avatar+91239 
0

A note to all guests:  :))

 

I wonder if the asker of this question will ever see the answer.

The asker draw my attention to it with a repost. The question is 2 days old not.

I guess it is not likely to be seen, what a pity.

 

This is a major reason why it is so much better to be a member.

If you are a member you can get email notifications that an answer has come in.

Your question also gets automatically stored in your watch list  and you can see notification there if you have a new answer.

Melody  Feb 24, 2017

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