I think u may have figure in the book.

1) To prove :ar(ΔBDE) =1/4 ar(ΔABC)

proof : we have given ABC is a equilateral triagle

so AB= BC = CA

we have also given that D is the mid point of BC
and BDE is a also an equilateral triangle

so BD = DE = EB

if u remember than we have studied heron't formula in which we have read that area of equilateral triangle = √3a²/4  where a is the side

so let the side of ΔABC = a
area = √3a²/4

and the side of ΔBDE = a/2 because D is the median of side of BC so BC  = a of 1/2 BC =a/2
and ar (ΔBDE) = √3/4 × (a/2)²
                         = √3a²/4 ×1/4 
                          = 1/4ar(ΔABC)     ( ∵√3a²/4 = ar(ΔABC)

the question you solved is not that one chapter is AREA RELATED TO CIRCLES -REVISION EXERCISE -Q NO.23