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In a three-dimensional space, the concept of equidistant points involves finding locations that maintain equal distances from specified objects. For instance, if we are considering two parallel planes, equidistant points from these planes would form another plane midway between them.

This new plane is unique because each of its points are equidistant from the two original planes, which means the distance from any point on this new plane to both of the original planes is identical. It's important to highlight that this scenario is straightforward only when the original planes are truly parallel.

If the two planes are not parallel, determining equidistant points becomes more complicated. In fact, in many cases, such points might not exist at all. That's because, as the angle or orientation between the planes changes, finding a consistent equidistance becomes infeasible.

Parallel planes play a crucial role in understanding three-dimensional geometry. They can be envisioned as endless and perfectly flat surfaces that never converge or intersect, no matter how far they extend.

• When two planes are parallel, they maintain a constant distance from one another.
• This constant distance enables us to easily interpret equidistant points, as these points form a third plane that is parallel and centered between the two original planes.

It's this arrangement that simplifies the geometry involved, making it easier to conceptualize how points relate to each other across space. When planes are not parallel, the complexity increases due to the lack of uniform distance, thus affecting how we perceive and calculate equidistant points.

The geometric representation of planes and points in 3D space is a key aspect of understanding spatial relationships. Visualizing these concepts can help in comprehending why certain statements, like the one in the exercise, are not universally applicable.

Imagining planes as vast sheets floating in space aids in visualizing equidistant points. When two planes are not parallel, their representation might intersect at a line, or they might never interact at all if skewed.

This inconsistency in their relative positions leads to the reality that finding a uniform set of equidistant points is possible primarily when dealing with parallel planes. Thus, geometric representation is a powerful tool aiding us in making sense of spatial relationships and understanding the logic behind various geometric truths or misconceptions.