Centroid, Incenter, Circumcenter, Orthocenter

Front

What is the definition of the centroid of a triangle?

Back

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

Front

If G is the centroid of triangle ABC, what is the relationship between the segments AG, GB, and GC?

Back

The centroid divides each median into two segments, with the ratio of the lengths being 2:1, where the longer segment is from the vertex to the centroid.

Front

What is the formula to find the length of a median in a triangle?

Back

The length of a median can be calculated using the formula: m = 1/2 * sqrt(2a^2 + 2b^2 - c^2), where a, b, and c are the lengths of the sides of the triangle.

Front

What is the definition of the incenter of a triangle?

Back

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

Front

How do you find the inradius of a triangle?

Back

The inradius (r) can be found using the formula: r = A/s, where A is the area of the triangle and s is the semi-perimeter.

Front

What is the circumcenter of a triangle?

Back

The circumcenter is the point where the three perpendicular bisectors of the sides of the triangle intersect, and it is the center of the circumscribed circle.

Front

What is the relationship between the circumcenter and the vertices of the triangle?

Back

The circumcenter is equidistant from all three vertices of the triangle.

Front

What is the definition of 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 orthocenter and the types of triangles?

Back

In an acute triangle, the orthocenter is inside the triangle; in a right triangle, it is at the vertex of the right angle; and in an obtuse triangle, it is outside the triangle.

Front

What is the significance of the angle bisector in a triangle?

Back

The angle bisector divides the angle into two equal parts and intersects the opposite side at a point that is proportional to the lengths of the other two sides.