Prove that if lengths of 2 medians of a triangle are equal then the triangle is isosceles ?

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Answer 1 · 2018-07-21T14:05:49

Drawn Drawn
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.