D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equilateral triangle ABC. Show that is also an equilateral triangle.
Solution.
D and E are midpoints of BC and AC respectively, so using mid-point theorem:
…..(1)
E and F are midpoints of AC and AB, so using mid-point theorem:
…..(2)
F and D are midpoints of AB and BC, so using mid-point theorem:
…..(3)
It is given that is an equilateral triangle
{dividing by 2}
Using 1, 2 and 3 we get
DE = EF = FD
Hence DEF is an equilateral triangle.
Hence proved