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 =
And
GF = ( As we know medians are trisected at centroid )
SO,
GE = = GF = ( Given BE = CF )
And
GC = = GB = ( 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 = And CE = ( 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 )
Given
Medians BE = CF , of a ABC ,
Let BE and CF meet at G the centroid .
In ECG And FBG
GE =
And
GF = ( As we know medians are trisected at centroid )
SO,
GE = = GF = ( Given BE = CF )
And
GC = = GB = ( 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 = And CE = ( 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 )