Medians BE and CF of a triangle ABC are equal. Prove that the triangle is isosceles

Answer :
Given
Medians BE  = CF , of a  ABC , 
Let BE and CF meet at G the centroid .

In  ECG And  ​FBG 

GE = BE3
And
GF = CF3    ( As we know medians are trisected at centroid  )
SO,

GE = BE3  = GF = CF3                    ( Given BE  = CF )

And
GC = 2CF3  = GB  = 2BE3    ( As we know medians are trisected at centroid and BE  = CF is given )
And
 EGC   =   FGB                    ( opposite angles )
Hence
  ECG  ​FBG                      ( By SAS rule )
So,
BF = CE                                     ( CPCT )
But BF  = AB2  And  CE  = AC2  ( As we know BE and CF are medians , So E and F are  the midpoints of AB and AC )
Hence
AB  = 2 BF = AC = 2CE
SO,
AB  = AC 
Hence
  ABC is a isosceles triangle                                     ( Hence proved )

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