G2.3 In Figure 1 , the square $P Q R S$ is inscribed in $\triangle A B C$. The areas of $\triangle A P Q, \triangle P B S$ and $\triangle Q R C$ are 4,4 and 12 respectively. If the area of the square is $c$, find the value of $c$.
Let $B C=a, P S=x$, the altitude from $A$ onto $B C=h$.
Area of $\triangle B P S=\frac{1}{2} x \cdot B S=4 \Rightarrow B S=\frac{8}{x}$
Area of $\triangle C Q R=\frac{1}{2} x \cdot C R=12 \Rightarrow C R=\frac{24}{x}$
$$
B C=B S+S R+R C=\frac{8}{x}+x+\frac{24}{x}=x+\frac{32}{x}
$$

Area of $\triangle A P Q=\frac{1}{2} x(h-x)=4 \Rightarrow h=\frac{8}{x}+x$.
Area of $\triangle A B C=\frac{1}{2} h \cdot B C=4+4+12+x^{2}=20+x^{2}$.
Sub. (1) and (2) into (3): $\frac{1}{2}\left(\frac{8}{x}+x\right) \cdot\left(x+\frac{32}{x}\right)=20+x^{2}$
$$
\begin{array}{l}
\left(8+x^{2}\right)\left(x^{2}+32\right)=40 x^{2}+2 x^{4} \\
x^{4}+40 x^{2}+256=40 x^{2}+2 x^{4} \\
x^{4}=256
\end{array}
$$
$c=$ area of the square $=x^{2}=16$