I5.3 In the figure, $L_{1}$ and $L_{2}$ are tangents to the three circles. If the radius of the largest circle is 18 and the radius of the smallest circle is $4 b$, find $c$, where $c$ is the radius of the circle $W$.
Let the centres of the 3 circles be $A, B, C$ as shown in the figure.
$L_{1}$ touches the circles at $D, E, F$ as shown.
$A D \perp L_{1}, W E \perp L_{1}, B F \perp L_{1}$ (tangent $\perp$ radius)
Let $A B$ intersects the circle $W$ at $P$ and $Q$.
$A D=A P=4 b=8, E W=W Q=P W=c$
$Q B=B F=18$ (radii of the circle)
Draw $A G / / D E, W H / / E F$ as shown
$E W / / F B$ (int. $\angle$ supp.)
$\angle A W G=\angle W B H$ (corr. $\angle$ s $E W / / F B$ )
$A G \perp G W, W H \perp H B$ (by construction)
$\triangle A G W \sim \triangle W H B$ (equiangular)
$G W=c-8, B H=c+18$ (opp. sides of rectangle)
$\frac{c-8}{c+8}=\frac{18-c}{c+18} \quad$ (ratio of sides, $\sim \Delta$ )
$(c-8)(c+18)=(c+8)(18-c)$
$c^{2}+10 c-144=-c^{2}+10 c+144$
$2 c^{2}=2(144)$
$c=12 \quad$