Point
GeometryA point is an exact location in space with no size, length, width, or depth, represented by a dot and named with a capital letter.
Definition
A point is an exact spot in space. It has no size at all - no length, no width, no thickness. We draw it as a tiny dot and name it with a capital letter like point $A$.
Example
The tip of a sharpened pencil is close to a point. On a map, the dot marking your city is like a point - it shows an exact location but has no actual size.
Key Insight
Everything in geometry starts with points. Lines, shapes, and all figures are built from points - making the point the most basic building block in all of geometry.
Definition
A point is a fundamental undefined term in geometry. It represents an exact location in space and has no dimension - no length, width, or depth. Points are named with capital letters and are used to define all other geometric figures.
Example
Points $A$ and $B$ are two distinct locations in a plane. A line passes through both points. A segment connects them. The coordinates $(3, 5)$ name a point on the coordinate plane.
Key Insight
Points are called "undefined terms" because they are so basic they cannot be defined using simpler geometric ideas. Instead, we accept what a point is intuitively and build all definitions on top of it.
Definition
In Euclidean geometry, a point is a primitive (undefined) notion satisfying the axioms of the system. In analytic geometry, a point in $\mathbb{R}^n$ is an ordered $n$-tuple of real numbers. In topology, a point is an element of a set on which a topology is defined.
Example
In $\mathbb{R}^2$, point $P = (x, y)$ identifies a unique location. The set of all points equidistant from a fixed point defines a circle. In projective geometry, points at infinity are added to make the system more uniform.
Key Insight
The choice to leave "point" undefined is deliberate: Euclid's definition ("that which has no part") is circular. Modern axiomatic systems (Hilbert, Tarski) treat point as a primitive satisfying stated axioms, avoiding the circularity.