91Ó°ÊÓ

To understand the exercise, it's key to know what a linear path is. In geometry, a linear path refers to the straight line connecting two points. Imagine points \( A \) and \( B \). If you draw a straight line from \( A \) to \( B \), you have created a linear path. This path will include every point that lies between \( A \) and \( B \).
Linear paths have some important properties:

• They have direction – from point \( A \) to point \( B \) or vice versa.
• They are the shortest distance between two points.
• All points on the path have coordinates that can be expressed as combinations of the coordinates of \( A \) and \( B \).

Since points \( A \) and \( B \) are in the plane \( \boldsymbol{P} \), the linear path connecting them is also in the plane \( \boldsymbol{P} \). So, if point \( C \) lies on this linear path, it must also be in plane \( \boldsymbol{P} \).

Collinearity is an essential concept in geometry. It means that three or more points lie on the same straight line. In our exercise, we have points \( A \), \( B \), and \( C \). Point \( C \) is on the line determined by \( A \) and \( B \), indicating that all three points are collinear.
The significance of collinearity:

• It tells us that these points share the same linear path.
• Any manipulation involving the line affects all collinear points similarly.

When dealing with planes, collinearity plays a key role. Since \( A \) and \( B \) are in plane \( \boldsymbol{P} \), their linear path also lies in this plane. Therefore, being collinear with \( A \) and \( B \) forces \( C \) to also be in plane \( \boldsymbol{P} \). This is a straightforward, but powerful visualization tool for understanding geometric relationships.

Plane geometry deals with figures on a flat surface called a plane. A plane extends infinitely in two dimensions: length and width. In this exercise, plane \( \boldsymbol{P} \) is mentioned, containing points \( A \) and \( B \).
Here are some key points about planes:

• Any line connecting two points in a plane lies entirely within that plane.
• A plane can be defined by three non-collinear points.
• All points that lie on a line in the plane will also lie in the plane.

When considering plane geometry, knowing whether a point lies within a given plane helps solve many geometric problems. Since points \( A \) and \( B \) are in plane \( \boldsymbol{P} \), the line determined by them is in \( \boldsymbol{P} \). Therefore, if point \( C \) lies on this line, it must be in plane \( \boldsymbol{P} \). Understanding these properties ensures you're comfortable imagining how lines and points interact within a plane, aiding in visualizing and solving related exercises.