Given a square of one unit side length. Four points are chosen on the square as shown. Then the quadrilateral EFGH so formed is a
Square
Using Phythagoras theorem in Δ AEH we get
EH=√104
Similarly we can get EH = FG = EF = GH
In ΔAEH and ΔHDG
AH = DG
AE = DH
∠ EAH = ∠ HDG (900)
Δ EAH ≅Δ HDG
Let ∠ AHE = x0
Then ∠ DGH = x0
∠ DHG = 900−x0 (Using sum of angles in a triangle is 1800)
∠EHG = 900 (using sum of angles on a straight line)
Similarly we can prove others also.
So EHGF is a square.