Two systems of rectangular axes have the same origin. If a plane cuts them at distances $a,b,c$ and $a^{\prime },b^{\prime },c^{\prime }$ from the origin, then:

The perpendicular distance $p$ from the origin $(0,0,0)$ to the plane is invariant regardless of the coordinate system chosen.
In the first system, the plane equation is $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$.
The distance $p$ is: $\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}$.
In the second system, the plane equation is $\frac{x^{\prime }}{a^{\prime }}+\frac{y^{\prime }}{b^{\prime }}+\frac{z^{\prime }}{c^{\prime }}=1$.
The distance $p$ is: $\frac{1}{p^2}=\frac{1}{a^{\prime 2}}+\frac{1}{b^{\prime 2}}+\frac{1}{c^{\prime 2}}$.
Equating the two:
$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{a^{\prime 2}}+\frac{1}{b^{\prime 2}}+\frac{1}{c^{\prime 2}}$$
$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}-\frac{1}{a^{\prime 2}}-\frac{1}{b^{\prime 2}}-\frac{1}{c^{\prime 2}}=0$$