Question

if D, E and F are the midpoints of the sides AB, BC and CA respectively of an isosceles triangle ABC prove that the triangle DEF is also isosceles.
from chapter midpoint theorem

Answer 1

Given: D,E,F are the mid points of AB,BC,CA
To Prove: DE = EF
Proof: AB = AC (isosceles triangle) 
          ∠B = ∠C (angle opp to equal sides)
          ¹/₂ AB = ¹/₂ AC
          so,DB = CF
          In ΔADF and ΔCEF
              DB = CF (proved above)
              BE = CE (E is the mid point of BC)
             ∠B = ∠C (proved above)
          ∴ ΔADF = ΔCEF (SAS)
               DE = FE (CPCT)
Hence, Proved

Answer 2

By using mid point theorem we can prove that triangle DEF is an isosceles triangle

Refer to the picture given along