Given:

There are two systems of rectangular axis.

Both the systems have same origin.

There is a plane that cuts both of these systems.

One system is cut at a distance of a, b, c.

The other system is cut at a distance of a’, b’, c’.

To Prove:

Proof: Since, a plane cuts both systems at distances a, b, c and a’, b’, c’ respectively. Then, this plane would have different equations in these two different systems, according to their distances with the system.

Let the equation of the plane in the system having distances a, b, c be

And, let the equation of the plane in the system having distances a’, b’, c’ be

According to the question,

The plane cuts the systems from the origin.

We know that,

Perpendicular distance of a plane ax + by + cz + d = 0 (where, not all a, b, c are zero) from the origin is given by

So,

Perpendicular distance of the plane from the origin is,

And,

Perpendicular distance of the plane from the origin is,

We also know that,

If two systems of lines have the same origin then their perpendicular distance from the origin to the plane in both the systems are equal.

So,

By cross multiplication,

Now, take square on both sides,

Or,

Hence, proved.