Circumcenter, Incenter, centroid, orthocenter

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 of the triangle.

Front

What is the incenter of a triangle?

Back

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

Front

What is the centroid of a triangle?

Back

The centroid is the point where the three medians of a triangle intersect. It divides each median into a ratio of 2:1.

Front

What is the orthocenter of a triangle?

Back

The orthocenter is the point where the altitudes of a triangle intersect. Its position varies depending on the type of triangle (acute, right, or obtuse).

Front

What are medians in a triangle?

Back

Medians are line segments that connect a vertex of the triangle to the midpoint of the opposite side.

Front

If BF = 10 in triangle ABC, what is AF if CF, BE, and AD are the medians?

Back

AF = 10, since the centroid divides each median in a 2:1 ratio.

Front

In triangle XYZ, if YC = 15 and XP = 8, what is PC?

Back

PC = 4, since the centroid divides the median in a 2:1 ratio.

Front

What is the relationship between the circumcenter and the circumradius?

Back

The circumradius is the radius of the circumcircle, which is the circle that passes through all three vertices of the triangle, centered at the circumcenter.

Front

How do you find the circumcenter of a triangle?

Back

To find the circumcenter, construct the perpendicular bisectors of at least two sides of the triangle. The point where they intersect is the circumcenter.

Front

What is the significance of the incenter in triangle geometry?

Back

The incenter is the center of the incircle, which is the largest circle that fits inside the triangle, tangent to all three sides.