Center (geometry)

The center of a circle or sphere is a point inside the circle which is equidistant from all points on the circle.

Triangle centers

• The centroid is where the three medians of the triangle meet.

• The incenter of the triangle is where the three angle bisectors meet. It is also the center of the incircle.

• The circumcenter is where the perpendicular bisectors of the triangles sides meet. It is also the center of the circumcircle.

• The orthocenter Is where the altitudes of the triangle meet.

Other notable centers include the nine-point center, symmedian point (Lemoine point), Nagel point, and Gergonne point.

Facts: The centroid, circumcenter, orthocenter, and nine-point center are always colinear on the Euler line.

The incenter of a triangle is the orthocenter of its excentral triangle, and the circumcenter of a triangle is the nine-point center of its excentral triangle, so the line connecting the circumcenter and incenter of a triangle is the Euler line of its excentral triangle.

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