একটি বাহু বা কর্ণ দেয়া থাকলে বর্গ অঙ্কন

A square is has a very nice, easy construction.

Given a square's side $AB$, draw two lines perpendicular to $AB$ through $A$ and through $B$. Then draw a two circles: one centered at $A$ through $B$, and one centered at $B$ through $A$. Where these circles intersect the perpendiculars through $A$ and through $B$, respectively, are the points $D$ and $C$, respectively.

Given a square's diagonal $AC$, the other diagonal must be the perpendicular bisector of $AC$. However, after constructing the perpendicular bisector, simply draw a circle with diameter $AC$ (centered at the intersection of the perpendicular bisector and through either $A$ or $C$) to find $B$ and $D$ along the perpendicular bisector.

Why are these true? Consider lengths, angles, and symmetry!