From Gogeometry

Using

• Pythagoras
• congruent triangles
• Similar triangles
• Equilateral triangle
• Area of a triangle
• Tangents to a circle
• Inradius in right triangle

•Pythagoras on triangle ΔBFC => (1) BF^2+CF^2=BC^2=a^2
•Incircle in a right triangle => (2) 2x=BF+CF-BC =BF+CF-a
•[ABCD]= 4 triangles areas + square with side 2x => (3) S=a^2=2BF.CF+4x^2
•(1 & 3) BF^2+CF^2=a^2=2BF.CF+4x^2
•BF^2+CF^2-2BF.CF=4x^2 •(CF-BF)^2=4x^2 => (4) CF-BF=2x
•(2 & 4) : 2x=BF+CF-a=CF-BF => (5) 2BF=a
•=> BF=a/2 and  BF ⊥ CF => ΔBFC is half of an equilateral triangle
•=> (6) CF=a√3/2 •(4 & 5 & 6) CF-BF=2x= a√3/2 – a/2
•=>x=(√3-1)a/4