Distance between points A and B is b.

    radius² = (a/2)² + (radius-x)²
    radius = x/2 + a²/8r

2.  You want to prove the Half-of-perimeter-to-radius ratio for circle is pi.

  A circle is a polygon with infinite number of side.

If the hexagon( sides = 6) above changed so that number of sides eventually become infinite then
the geometry becomes a circle.

 Ø = 360°/n
 x = r * sin(Ø/2)
side = 2 * x = 2*r*sin(Ø/2) = 2*r*sin(180°/n)

Perimeter = n * side = 2*n*r*sin(180°/n)

Perimeter-to-radius ratio = 2*n*sin(180°/n)

When n =3
Half-of-Perimeter-to-radius-ratio = 3*sin(180°/3) = 6/sqrt(3) = 2.5980

when n = 4(square)
Half-of-Perimeter-to-radius-ratio = 4*sin(180°/4) = 4/sqrt(2) = 2.8284

when n = 32
Half-of-Perimeter-to-radius-ratio = 32*sin(180°/32) = 3.1365

when n = infinity
 Limit             n*sin(180°/n)
 n->infinity

Applying following rule:
u(x) * v(x) = u´(x) * v(x) + v´(x) * u(x)

Half-of-Perimeter-to-radius-ratio =
Limit             n*cos(180°/n)*180°/n + 1 * sin(180°/n)  =  1*180°+0 = pi
 n->infinity