Circumcenter/Incenter

Front

What is the definition of 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 definition of 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

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

How do you find the incenter of a triangle?

Back

To find the incenter, construct the angle bisectors of at least two angles of the triangle. The point where they intersect is the incenter.

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. The circumcenter is the center of this circle.

Front

What is the relationship between the incenter and the inradius?

Back

The inradius is the radius of the incircle, which is the circle that is tangent to all three sides of the triangle. The incenter is the center of this circle.

Front

If a triangle has vertices A, B, and C, how can you find the circumcenter using coordinates?

Back

To find the circumcenter using coordinates, calculate the midpoints of two sides, find the slopes of those sides, determine the slopes of the perpendicular bisectors, and solve the equations of the lines to find their intersection.

Front

What is the significance of the circumcenter in triangle geometry?

Back

The circumcenter is significant because it helps in constructing the circumcircle and is used in various geometric proofs and constructions.

Front

What is the significance of the incenter in triangle geometry?

Back

The incenter is significant because it helps in constructing the incircle and is used in various geometric proofs and constructions.

Front

In a right triangle, where is the circumcenter located?

Back

In a right triangle, the circumcenter is located at the midpoint of the hypotenuse.