Question

show that the quadrilateral formed by joining the mid points of the adjacent sides of a square, is also a square

Answer 1

In the figure,

AF = AE = ED = DH = CH = CG = BG = BF

Therefore,
AE2 + AF2 = ED2 + DH2 = HC2 + GC2 = GB2+ BF2

By Pythagoras theorem,
EF2 = EH2 = HG2 = GF2
EF = HE = GH = GF
Hence all sides are equal.
EFGH is a square.