a

b

c

d

r

a^2+b^2+c^2+d^2=4r^2

PQ=d, QS=b

By symmetry,

RS=PQ=d.

CD=b-d

(This is the intersecting chords theorem.)

ac=bd

So

Angles ASP and ADP are equal by the circle theorems.

Therefore triangles AQS and PQD are similar and so 

Applying Pythagoras's Theorem to triangle ABD:

(a+c)^2+(b-d)^2=(2r)^2

Since

ac=bd

it follows that

A_{1} = (3.3218181818181813, 1)
B_{1} = (8.832193178850265, 4.039537355165376)
c = Circle(A_{1}, B_{1})
D = Intersect(c, yAxis, 1)
P = Intersect(c, xAxis, 1)
S = Intersect(c, xAxis, 2)
A = Intersect(c, yAxis, 2)
G = Intersect(xAxis, yAxis)
f = Segment(A, D)
g = Segment(P, S)
h = Segment(A_{1}, B_{1})
α = Angle(g, f)
β = Angle(f, g)
i = OrthogonalLine(A, f)
j = OrthogonalLine(D, f)
B = Intersect(c, i, 2)
k = OrthogonalLine(B, i)
C = Intersect(c, k, 2)
l = Segment(B, D)
m = Segment(A, B)
n = Segment(B, C)
p = Segment(C, D)
Q = Intersect(f, g)
R = Intersect(g, n)