Question

prove that a quadrilateral obtained by the mid point of sides of a square ,is also a square

Answer 1

Let a square ABCD in which L,M,N&O are the midpoints .

in triangle AML and triangle CNO 
       AM = CN ( AB = DC and M and O are the midpoints )
       AL = CM ( AD = BC and L and   N are the midpoints )
       angle MAL = angle NCO ( all angles of a square = 90 degree )
      by AAS critaria
        triangle AML CONGRUENT to triangle CNO
    therefore ML = ON  ( CPCT  )
similarly in triangle MBN CONGRUENT to  LDO  and 
    AND triangle  AML is CONGRUENT  to triangle
now , 
  in Triangle AML ,
 angle AML = angle ALM ( AM = AL ) 
                   = 45 degree
   similarly in triangle LDO 
   angle DLO = 45 degree
 there fore ,
 angle MLO = 90 degree 

by the properties of SQUARE 
 all sides are equal and angles are 90 degree