If $x_1,x_2,x_3$ and $y_1,y_2,y_3$ are both in G.P. with the same common ratio, then the points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$:

Let the common ratio be $r$.$x_2=x_{1}r,x_3=x_1{r}^2$$y_2=y_{1}r,y_3=y_1{r}^2$ Calculate the slope of the line segment joining $(x_1,y_1)$ and $(x_2,y_2)$:
$$m_{12}=\frac{y_2-y_1}{x_2-x_1}=\frac{y_{1}r-y_1}{x_{1}r-x_1}=\frac{y_1(r-1)}{x_1(r-1)}=\frac{y_1}{x_1}$$
Calculate the slope of the line segment joining $(x_2,y_2)$ and $(x_3,y_3)$:
$$m_{23}=\frac{y_3-y_2}{x_3-x_2}=\frac{y_1{r}^2-y_{1}r}{x_1{r}^2-x_{1}r}=\frac{y_{1}r(r-1)}{x_{1}r(r-1)}=\frac{y_1}{x_1}$$
Since the slopes are equal, the points are collinear and lie on a straight line.