The incentre - except in an equilateral triangle where it coincides with the centroid (for example).
Centroid
Centroid
The point where the altitudes of a triangle intersect is called the orthocenter. This point is concurrent, meaning the three altitudes intersect at this single point inside or outside the triangle. The orthocenter is different from the centroid, circumcenter, and incenter of a triangle.
The medians of a triangle are concurrent at a point called the centroid.
The bisectors of the angles of a triangle are concurrent at a point called the incentre which is also the centre of the inscribed circle that touches all three sides.
circumcenter
Angle bisectors intersect at the incenter which is equidistant from the sides
The altitudes of a triangle intersect at a point called the Orthocentre.Note : This is often stated as, "The altitudes are concurrent at a point called the Orthocentre."
Concurrent lines are three or more lines that intersect at a single point in a plane. This point of intersection is known as the point of concurrency. Concurrent lines are often discussed in geometry, particularly in the context of triangles, where important points like the centroid, orthocenter, and circumcenter are formed by concurrent lines drawn from the triangle's vertices or sides.
The medians of a triangle are concurrent and the point of concurrence, the centroid, is one-third of the distance from the opposite side to the vertex along the median
Any triangle has 3 medians Another answer (depending on what you are looking for) is that a triangle has concurrent medians (which means all three medians intersect at a single point).
To find the incenter of a triangle, which is the point where the angle bisectors of the triangle intersect, you need to construct the angle bisectors of at least two of the triangle's angles. Concurrent constructions involve drawing the angle bisectors using a compass and straightedge, ensuring they meet at a single point. This point is the incenter, equidistant from all three sides of the triangle. Additionally, constructing the incircle can further confirm the incenter's position.