**Given**

In #Delta ABC#, D and E are mid points of AB and AC.

Two medians # BE=CD#.

**RTP:** #Delta ABC# is isosceles.

**Construction**

#D,E# are joined. #DX and EY# are two perpendiculars drawn on BC.

**Proof**

#D and E# being mid points of #AB and AC# , the line #DE# must be parallel to #BC#. Hence perpendiculars #DX=EY#

Now in #DeltaCDXand Delta BEY#

#angle BYE=angleCXD=90^@#

# BE=CD#. given

and

#DX=EY# proved

#DeltaCDX~=Delta BEY# following #RHS# rule.

So

#angle EBY=angleDCXor angle EBC=angleDCB#

Now in #DeltaDCBand DeltaEBC# we have

# BE=CD#.given

#BC# common

and

#angle EBC=angleDCB#

So #DeltaDCB~= DeltaEBC# by SAS rule

So #angleDBC= angleECB#

#=>angleABC= angleACB#

This means #Delta ABC# is isosceles.