G1.4 Three circles of radii 2,3 and 10 units are placed inside another big circle in such a way that all circles are touching one another. Determine the value of the radius of the big circle.
Let $A$ be the centre of circle $C_{1}$ with radius $10, B$ be the centre of circle $C_{2}$ with radius $3, C$ be the centre of circle $C_{3}$ with radius 2 . Join $A B, B C, A C$.
Suppose $C_{1}$ and $C_{2}$ touch each other at $D, C_{2}$ and $C_{3}$ touch each other at $E, C_{3}$ and $C_{1}$ touch each other at $F$. Then $A, D, B$ are collinear, $B, E, C$ are collinear, $C, F, A$ are collinear.
$$
\begin{array}{l}
A B=10+3=13, B C=3+2=5, A C=10+2=12 \\
B C^{2}+A C^{2}=5^{2}+12^{2}=25+144=169=13^{2}=A B^{2}
\end{array}
$$
$\therefore \angle A C B=90^{\circ}$ (converse, Pythagoras' theorem)
Let $O$ be the centre of circle $C_{4}$ with radius $r$ circumscribing all three circles $C_{1}, C_{2}, C_{3}$ at $G, H$ and $I$ respectively. Then $O, A, G$ are collinear, $O, B, H$ are collinear, $O, C, I$ are collinear.
$$
A G=10, B H=3, C I=2, O A=r-10, O B=r-3, O C=r-2 \text {. }
$$

Let $P$ and $Q$ be the feet of perpendiculars drawn from $O$ onto $A C$ and $A B$ respectively.
Then $O P C Q$ is a rectangle.
Let $C P=x=Q O$ (opp. sides of rectangle), $C Q=y=P O$ (opp. sides of rectangle) $A P=12-x, B Q=5-y$.
In $\triangle O C P, x^{2}+y^{2}=(r-2)^{2}$
(1) (Pythagoras' theorem)

In $\triangle O A P,(12-x)^{2}+y^{2}=(r-10)^{2}$
(2) (Pythagoras' theorem)

In $\triangle O B Q, x^{2}+(5-y)^{2}=(r-3)^{2}$
(3) (Pythagoras' theorem)
(1) $-(2): 24 x-144=16 r-96 \Rightarrow x=\frac{2 r+6}{3} \cdots \cdots$
(1) $-(3): 10 y-25=2 r-5 \Rightarrow y=\frac{r+10}{5} \cdots \cdots$

Sub. (4) and (5) into (1): $\left(\frac{2 r+6}{3}\right)^{2}+\left(\frac{r+10}{5}\right)^{2}=(r-2)^{2}$
$$
\begin{array}{l}
25\left(4 r^{2}+24 r+36\right)+9\left(r^{2}+20 r+100\right)=225\left(r^{2}-4 r+4\right) \\
116 r^{2}-1680 r-900=0 \Rightarrow 29 r^{2}-420 r-225=0
\end{array}
$$
$(r-15)(29 r+15)=0 \Rightarrow r=15$, the radius of the big circle is 15 .