Question

In what type(s) of triangle can a vertex of the

triangle be one of the points of concurrency of the triangle? Explain.

Answer 1

In geometry, a vertex of a triangle can be a point of concurrency in a right-angled triangle, where the orthocenter is at the right angle vertex, and an obtuse triangle, where the orthocenter can coincide with a vertex if it is extended.

Step-by-step explanation:

In mathematics, specifically in geometry, the vertices of certain triangles can be points of concurrency. A point of concurrency is the point where three or more lines intersect. In the case of triangles, there are four classic points of concurrency: the centroid, the circumcenter, the orthocenter, and the incenter.

The orthocenter is the point of concurrency that can lie exactly at a triangle's vertex. This occurs in two types of triangles:

• A right-angled triangle, where the orthocenter is at the vertex forming the right angle because the altitudes are the sides of the triangle itself.
• An obtuse triangle, where the orthocenter is outside the triangle, but it may lie at the extension of a side, thus coinciding with a vertex if extended.

In other types of triangles, like acute and equilateral, the orthocenter falls inside the triangle but does not coincide with any vertex. For the centroid, circumcenter, and incenter, these points of concurrency do not typically coincide with a vertex.