Descartes Sangaku : The Beast part 3

Let the radius of the red circle be $R$ . Then, applying Descartes's circle's theorem we get

$\dfrac 1R=\dfrac 1a+\dfrac {1}{1-a}-1+2\sqrt {\dfrac {1}{a(1-a)}-\dfrac 1a-\dfrac {1}{1-a}}$

$=\dfrac {a^2-a+1}{a(1-a)}$

where $a$ and $1-a$ are the radii of the cyan and the green semicircles respectively.

And,

$17=\dfrac 1a+\dfrac {1}{1-a}+\dfrac {a^2-a+1}{a(1-a)}+2\sqrt {\dfrac {1}{a(1-a)}+\dfrac {a^2-a+1}{a(1-a)^2}+\dfrac {a^2-a+1}{a^2(1-a)}}$

$=\dfrac {a^2-a+4}{a-a^2}$

$\implies 9a^2-9a+2=0\implies a=\dfrac 13,1-a=\dfrac 23$

$\implies R=\dfrac 27$

Total area of the cyan and green semicircles, red circle and black circle is

$\dfrac π2\left (\dfrac 19+\dfrac 49+\dfrac {8}{49}+\dfrac {2}{289}\right )$

$=\dfrac π2\times \dfrac {92495}{127449}$

Hence area of the white region is

$\dfrac π2\left (1-\dfrac {92495}{127449}\right ) =π\times \dfrac {17477}{127449}$

Therefore $a=17477,b=127449,b-a=\boxed {109972}$ .

• To solve this problem, we shall denote the radius of the cyan semi-circle as $\boxed{1-x}$ , the radius of the green semi-circle as $\boxed{x}$ and the radius of the red circle as $\boxed{R(x)}$ .

• We shall use the Pythagorean theorem to express $\boxed{R(x)}$ in terms of $\boxed{x}$

• We shall write three equations using the diagram:

• $\boxed{(1-R(x))^{2}=l(x)^{2}+h(x)^{2}}$
• $\boxed{(1-x+R(x))^{2}=(x-l(x))^{2}+h(x)^{2}}$
• $\boxed{(x+R(x))^{2}=(1-x+l(x))^{2}+h(x)^{2}}$

---

• After combining the equation and using minor algebra skills we find:
• $\boxed{R(x)=\frac{x-x^{2}}{x^{2}-x+1}}$

---

• Now using Descartes Circle's theorem whe can express the radius of the black circle in terms of $x$ :
• $\boxed{\frac{1}{B\left(x\right)\:}=\frac{1}{R\left(x\right)}+\frac{1}{x}+\frac{1}{1-x}+2\sqrt{\frac{1}{R\left(x\right)\cdot x}+\frac{1}{R\left(x\right)\cdot \left(1-x\right)}+\frac{1}{x\cdot \left(1-x\right)}}}$
• Whe can solve then $\boxed{\frac{1}{B(x)}=17}$
• We find $\boxed{x=\frac{2}{3}}$
• We evaluate : $\boxed{R(\frac{2}{3})=\frac{2}{7}}$

---

• Knowing all radii needed when find that the white area is : $\boxed{\pi \cdot \frac{17477}{127449}}$
• $\boxed{127449-17447=109972}$