G3.3 $A B C D$ is a cyclic quadrilateral. $A C$ and $B D$ intersect at $G$. Suppose $A C=16 \mathrm{~cm}, B C=C D=8 \mathrm{~cm}$, $B G=x \mathrm{~cm}$ and $G D=y \mathrm{~cm}$. If $x$ and $y$ are integers and $x+y=c$, find the value of $c$.
As shown in the figure, let $C G=t, A G=16-t$.
Let $\angle C B G=\theta, \angle A C B=\alpha$.
Then $\angle C A B=\theta$ (eq. chords eq. $\angle$ s)
Then $\triangle B C G \sim \triangle A C B$ (equiangular)
$t: 8=8: 16$ (ratio of sides, $\sim \Delta \mathrm{s}$ )
$t=4$
It is easy to see that $\triangle A D G \sim \triangle B C G$ (equiangular)
$$
\begin{array}{l}
(16-t): y=x: t \text { (ratio of sides, } \sim \Delta \mathrm{s} \text { ) } \\
(16-4) \times 4=x y \\
x y=48
\end{array}
$$

Assume that $x$ and $y$ are integers, then possible pairs of $(x, y)$ are $(1,48),(2,24), \ldots .,(6,8), \ldots,(48,1)$.
Using triangle inequality $x+t>8$ and $8+t>x$ in $\triangle B C G$, the only possible combinations are:
$$
\begin{array}{l}
(x, y)=(6,8) \text { or }(8,6) \\
c=x+y=14
\end{array}
$$