Newtonian Form of the Thin-Lens Formula

The formula $\frac{1}{p}+\frac{1}{i}=\frac{1}{f}$ is called the Gaussian form of the thin-lens formula. another form of this formula, the Newtonian form, is obtained by considering the distance $x$ from the object to the first focal point and the distance $x'$ from the second focal point to the image. I am going to show that $xx'=f^{2}$ is the Newtonian form of the thin-lens formula.

In this situation, my diagram is going to look as follows below:

Starting with the thin lens formula, $\frac{1}{p}+\frac{1}{i}=\frac{1}{f}$, I note that the object distance is $x+f$, and the image distance is $x'+f$, where the focal distances for the two lenses are equivalent. After making these substitutions into the thin lens formula, and solving for $f$ I obtain the equation $f=\frac{xx'+fx+fx'+f^{2}}{x'+x+2f}$. Now I can multiply each side of the equation by $x'+x+2f$ and solve for $f^{2}$  to obtain the final solution $f^{2}=xx'$, which is the Newtonian form of the thin lens equation.