In the context of geometry, two points are said to be in the same plane if they lie on the same flat surface. This is a fundamental concept in Euclidean geometry, where all points in a plane are equidistant from a common point known as the center, typically the origin. The idea of a plane is crucial in understanding spatial relationships and is foundational to various fields, including architecture, engineering, and physics.

The reason two points are always in the same plane is due to the nature of Euclidean space. In a three-dimensional space, a plane is defined by any three non-collinear points, meaning points that do not lie on the same straight line. Once a plane is defined by three points, any two points that are not collinear with these three points will also lie on the same plane. This is because the plane is a two-dimensional surface that extends infinitely in all directions, encompassing all points that are not on the line connecting the three defining points.

According to the Euclidean postulate, through any two points, there is exactly one line. Extending this to the concept of a plane, through any two points, there is exactly one plane. This is why any two points, regardless of their position in space, will always be in the same plane. The data supporting this concept is derived from the axioms and postulates of Euclidean geometry, which have been widely accepted and used in mathematical and scientific applications for centuries.

For instance, in architectural design, understanding that any two points are in the same plane is essential for creating accurate floor plans and spatial layouts. Similarly, in physics, the concept of a plane is used to describe the motion of objects in two-dimensional space, such as the trajectory of a projectile or the path of a planet in its orbit. The simplicity and universality of this concept make it a cornerstone of geometric understanding and practical application across various disciplines.