P, Q and R are respectively the mid-points of sides BC, AC and AB of an equilateral triangleABC. prove that trianglePQR is also an equilateral triangle.

Answer:

We have equilateral triangle , And have mid points P , Q and R on sides BC , AC and AB respectively , As:
                             
We know In $∆$ ABC
AB = BC = AC 
And
AR = RB = BP = PC = CQ = QA
In ​ $∆$ BRP , we know
$\angle$ ABC = 60$°$                                 ( As angle of equilateral ​$∆$ ABC )
$\angle$ BPR = ​$\angle$ BRP                              ( opposite angle of equal sides as RB = BP )
And 
$\angle$ ABC + $\angle$ BPR + ​ ​$\angle$ BRP  = 180$°$ 
$\Rightarrow$ ​60$°$   ​+  $\angle$ BPR + ​ ​$\angle$ BRP = 180$°$ 
$\Rightarrow$  2  $\angle$ BPR = 120$°$                     ( AS:$\angle$ BPR = ​$\angle$ BRP  )
$\Rightarrow$   $\angle$ BPR​  = 60$°$ 
Therefore
$\angle$ ABC =  $\angle$ BPR =  ​$\angle$ BRP  = ​60$°$ 
So ​ $∆$ BRP is also a equilateral triangle , So
RB = BP = RP
Similarly we can show
PC = CQ = PQ
And,
AR ​= QA = QR
Therefore

RP = PQ = QR   , Thats shows $∆$ PQR is equilateral triangle .              ( Hence proved )