Question

D ,E and F are respectively the mid points of sides BC, CA and AB respectively of an equilateral triangle ABC.show that ∆ DEF is also an equilateral triangle.

Answer 1

 Given :

An equilateral triangle ABC which D, E, F are the mid-points of BC, CA and AB respectively.

 
 
To prove :
triangleDEF is also equilateral.

 
 
PROOF :
In triangleABC,
 
F and E are the mid-points of AB and AC,
FE // BC & ----{1} [midpoint theorom]
 
FE = 1/2BC [midpoint theorom]
 
ie, FE = BD ----{2}
 
From 1 and 2,
 
In quadrilateral FEDB, one pair of opposite sides are equal and parallel,
FEDB is a parallellogram.
Therefore, angleFBD = angleDEF [opposite angles of a parallelogram are equal]
 
since, angleFBD = 60degree [ABC is an equilateral triangle]
angleDEF = 60degree [opposite angles of a parallelogram are equal]
 
 
 
Similarlly,
FECD is a parallelogram,
 
since, angleECD = 60degree [ABC is an equilateral triangle]
angleDFE = 60degree [opposite angles of a parallelogram are equal]
 
 
Similarlly,
FDEA s a parallelogram,
since, angleFAE = 60degree [ABC is an equilateral triangle]
angleEDF = 60degree [opposite angles of a parallelogram are equal]
 
 
Since, all the angles in trianlgeDEF is 60degree ,
triangleDEF is an equilateral triangle.
 
        HENCE PROVED      

Answer 2

it is very easy way
hope you understood