Given: In equilateral ΔABC, D, E and F are the mid-points of sides BC, CA and AB, respectively.
To show ΔDEF is an equilateral triangle.
Proof: Since in ΔABC, E and F are the mid-points of AC and AB respectively, then EF || BC and
EF =½ BC ,,.(i)
DF || AC, DE || AB
DE = ½ AB and FD = ½ AC [by mid-point theorem]… (ii)
since ΔABC is an equilateral triangle
AB = BC = CA
=> ½ AB = ½ BC = ½ CA [dividing by 2]
=> ∴ DE = EF = FD [from Eqs. (i) and (ii)]
Thus, all sides of ADEF are equal.
Hence, ΔDEF is an equilateral triangle.
Hence proved.
