Centers of Triangles

Front

What is the definition of the incenter of a triangle?

Back

The incenter of a triangle is the point where the angle bisectors of the triangle intersect. It is equidistant from all three sides of the triangle.

Front

What is the definition of a median in a triangle?

Back

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.

Front

What is the definition of a perpendicular bisector in a triangle?

Back

A perpendicular bisector of a side of a triangle is a line that is perpendicular to that side and passes through its midpoint.

Front

How many medians does a triangle have?

Back

A triangle has three medians, one from each vertex.

Front

What is the centroid of a triangle?

Back

The centroid is the point where the three medians of a triangle intersect. It is also the center of mass of the triangle.

Front

What is the circumcenter of a triangle?

Back

The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect. It is equidistant from all three vertices.

Front

What is the orthocenter of a triangle?

Back

The orthocenter is the point where the three altitudes of a triangle intersect.

Front

What is the relationship between the incenter and the circumradius?

Back

The incenter is the center of the inscribed circle (incircle) of the triangle, while the circumradius is the radius of the circumscribed circle (circumcircle) around the triangle.

Front

What is the significance of the centroid in a triangle?

Back

The centroid divides each median into two segments, with the segment connecting the vertex to the centroid being twice as long as the segment connecting the centroid to the midpoint of the opposite side.

Front

What is the relationship between the orthocenter and the circumcenter in an acute triangle?

Back

In an acute triangle, the orthocenter lies inside the triangle, while the circumcenter lies outside the triangle.