What are the 4 points of concurrency?

The term points of concurrency refers to specific points in geometry where multiple lines, such as medians, altitudes, or perpendicular bisectors, intersect. There are four main points of concurrency commonly studied in triangles:

1. Centroid

• Definition: The centroid is the point where the three medians of a triangle intersect. A median is a line drawn from a vertex to the midpoint of the opposite side.
• Properties: The centroid is often referred to as the "center of gravity" or "balance point" of the triangle because it divides each median into two segments, with the longer segment being twice the length of the shorter one.

2. Circumcenter

• Definition: The circumcenter is the point where the three perpendicular bisectors of the sides of a triangle meet. A perpendicular bisector is a line that is perpendicular to a side of the triangle and divides it into two equal parts.
• Properties: The circumcenter is equidistant from all three vertices of the triangle, which makes it the center of the circumcircle (a circle that passes through all three vertices of the triangle). The circumcenter can lie inside, on, or outside the triangle, depending on whether the triangle is acute, right, or obtuse, respectively.

3. Incenter

• Definition: The incenter is the point where the three angle bisectors of a triangle intersect. An angle bisector divides an angle into two equal angles.
• Properties: The incenter is equidistant from all three sides of the triangle and serves as the center of the incircle (a circle that is tangent to all three sides of the triangle). The incenter always lies inside the triangle.

4. Orthocenter

• Definition: The orthocenter is the point where the three altitudes of a triangle intersect. An altitude is a perpendicular line drawn from a vertex to the opposite side (or its extension).
• Properties: The orthocenter can be located inside, on, or outside the triangle, depending on whether the triangle is acute, right, or obtuse. For right triangles, the orthocenter is at the vertex of the right angle.

Conclusion

These four points of concurrency—centroid, circumcenter, incenter, and orthocenter—are critical in understanding the properties and relationships within triangles in geometry. Each point offers unique characteristics related to the triangle's sides and angles, providing valuable insights into geometric constructions and theorems.