Geometry - Incenter, Circumcenter, Orthocenter, Centroids

What is the incenter of a triangle?

The incenter is the point where the angle bisectors of a triangle intersect, and it is also the center of the circle inscribed within the triangle.

What is the circumcenter of a triangle?

The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect, and it is the center of the circumcircle that passes through all three vertices of the triangle.

What is the orthocenter of a triangle?

The orthocenter is the point where the altitudes of a triangle intersect, and its location can vary; it lies inside the triangle for acute triangles, on the triangle for right triangles, and outside for obtuse triangles.

What is the centroid of a triangle?

The centroid is the point where the three medians of a triangle intersect, and it represents the triangle's center of mass; it is also the balance point of the triangle.

What is the incenter of a triangle, and how can it be constructed?

The incenter is the point where the angle bisectors of a triangle intersect. It can be constructed by drawing the angle bisectors of each of the triangle's three angles.

What is the circumcenter of a triangle, and how can it be determined?

The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect. It can be determined by constructing the perpendicular bisectors of at least two sides of the triangle.

What is the orthocenter of a triangle, and how does its position vary?

The orthocenter is the point where the altitudes of a triangle intersect. Its position varies: it lies inside the triangle for acute triangles, on the triangle for right triangles, and outside for obtuse triangles.

What is the centroid of a triangle, and what are its properties?

The centroid is the point where the three medians of a triangle intersect. It acts as the triangle's center of mass and divides each median into a segment ratio of 2:1.