Inspired By Islamic Art

Right triangle $\Delta ABC$ shows the geometrical relation between all $5$ circles, the radii being $1, a, b, c, d$ from smallest to largest. The following relations must hold

$c=1+\sqrt { 2 }$

$1+c+\sqrt { { \left( b+c \right) }^{ 2 }-{ b }^{ 2 } } =2b+d$

$\sqrt { 2 } \left( 2b+d \right) =\dfrac{1}{\sqrt{2}}(1+c)+\sqrt { { \left( a+c \right) }^{ 2 }-{ c }^{ 2 } } +a+d$

$\sqrt { { \left( a+d \right) }^{ 2 }+{ \left( b+d \right) }^{ 2 }-2(a+d)(b+d)Cos\left( 45 \right) } =a+b$

which numerically leads to

$a=1.701800…$
$b=2.180149…$
$c=2.414235…$
$d=3.098061…$

The area of the triangle, the circles inside the triangle, and the ratio are

$\Delta ABC=\dfrac { 1 }{ 2 } { \left( 2b+d \right) }^{ 2 }=27.813599...$

Circles $=\dfrac { 1 }{ 2 } \pi { \left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } \right) +\dfrac { 1 }{ 8 } \pi \left( 1+{ d }^{ 2 } \right) }=25.332378...$

Ratio $=0.910792…$

Here's a sketch copy of the original Islamic artwork (which I can't find now) that suggested this problem