Point (Geometry)

In geometry, a point is a primitive notion that is not formally defined, serving as the foundation for constructing all other geometric concepts.

Overview of a Point

Geometric nature: A point is understood as a spatial object with zero dimensions in all directions (a zero-dimensional object, $0\text{D}$ ).
Visual representation: A tiny dot can be considered a visual representation of a point; hence, a point is conventionally depicted by a dot symbol $(\bullet)$ .
Naming conventions: A point is usually denoted by an uppercase Latin letter such as $A, B, C, M, N\dots$ , or less commonly by Greek letters.
Relationship with other geometric figures: A point is itself a geometric figure. Every curve or line is a set consisting of infinitely many points:
- Line segments, rays, and straight lines are sets of collinear points.
- A circle is the set of all points equidistant (by the radius) from a fixed point (the center).
- Conic sections (ellipse, parabola, hyperbola) are sets of points satisfying specific distance conditions.

Intersection point: A point belonging to two or more distinct lines or curves.
Origin (Endpoint of a ray): The terminating boundary point of a half-line (or ray).
Point of tangency: The single common point where a tangent line (or plane) touches a curve (or curved surface).
Endpoints: The two terminating points of a line segment that bound the segment.
Midpoint: A point lying on a line segment that is equidistant from both endpoints of that segment.
Vertex (polygon, polyhedron): The common points shared by the edges of a polygon or the faces of a polyhedron.
Center of a circle: The point equidistant from all points situated on that circle.
Foci of an ellipse: The two fixed points such that the sum of the distances from them to any point on the ellipse is constant.

Centroid: The point of concurrency of the three medians of a triangle.
Orthocenter: The point of concurrency of the three altitudes of a triangle.
Circumcenter: The intersection of the three perpendicular bisectors, equidistant from the three vertices.
Incenter: The intersection of the three internal angle bisectors, equidistant from the three sides.
Excenter: The intersection of two external angle bisectors and one internal angle bisector of the opposite angle.
Symmedian point (Lemoine point): The point of concurrency of the three symmedians in a triangle.
Brocard points: Special interior points of a triangle where the directed angles formed with the vertices are equal.
Euler center (Nine-point center): The center of the nine-point circle which passes through the side midpoints, altitude feet, and Euler points.