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

Explanation

Given In equilateral `Delta`ABC, D, E and F are the mid-points of sides BC, CA and AB, respectively. ltBrgt To show `Delta`DEF is an equilateral triangle.

Proof Since in `Delta`ABC, E and F are the mid-points of AC and AB respectively, then EF`||`BC and
`" "EF=(1)/(2)BC" "...(i)`
Similarly, `" "DF||AC, DE||AB`
and `" "DE=(1)/(2)AB and FD =(1)/(2)AC ` [by mid-point theorem]...(ii)
since `Delta`ABC is an equilateral triangle ltBrgt `therefore" "AB=BC=CA`
`rArr" "(1)/(2)AB=(1)/(2)BC=(1)/(2)CA" "`[dividing by 2]
`rArr" "DE=EF=FD" "` [from Eqs. (i) and (ii) ]
Thus, all sides of `Delta`DEF are equal.
Hence, `Delta`DEF is an equilateral triangle. `" "` Hence proved.