$\begin{array}{l}\text{ If }(a,b),(c,d),(e,f)\text{ are the vertices of a triangle such that }a\text{ , }c,e\text{ are in G.P. with common ratio }r\text{ and }b,d,f\text{ are in G.P. with }\\ \text{ common ratio }s\text{ then the area of the triangle is }\end{array}$

$\begin{array}{c}\text{Here }c=ar,e=a{r}^2,d=bs,f=b{s}^2\\ \text{The area of triangle,}A=\frac{1}{2}\left|\begin{array}{lll}a& b& 1\\ c& d& 1\\ c& f& 1\end{array}\right|\\ =\frac{1}{2}\left[\right(\text{cf}-\text{ed})+(\text{be}-\text{af})+(\text{ad}-\text{ac}\left)\right]\\ =\frac{1}{2}\left[{\text{abrs}}^2-{\text{abr}}^2 \text{s}+{\text{abr}}^2-{\text{abs}}^2+\text{abs}-\text{bar}\right]\\ =\frac{1}{2}\text{ab}\left[\text{rs}(\text{s}-\text{r})+\left({\text{r}}^2-{\text{s}}^2\right)+(\text{s}-\text{r})\right.\\ =\frac{1}{2}ab(s-r)[rs-s-r+1]\\ =\frac{1}{2}\text{ab}(\text{s}-\text{r})(\text{s}-1)(\text{r}-1)\end{array}$