Triangle centers

Triangle centers are distinguished points defined by a triangle’s sides, angles, or associated constructions. Examples include the centroid, incenter, circumcenter, and orthocenter.

Angle bisector: A ray or line that divides an angle into two equal angles. The three internal angle bisectors meet at the incenter.
Centroid: The point where a triangle’s three medians intersect; it is their common balance point. It is the center obtained from vertex-to-midpoint constructions.
Ceva's theorem: A theorem giving a product condition for three lines through a triangle’s vertices to be concurrent. It can establish concurrency for lines defining triangle centers.
Acute triangle: A triangle whose three interior angles are less than 90 degrees. Its circumcenter and orthocenter both lie inside the triangle.
Median (geometry): A segment joining a triangle’s vertex to the midpoint of the opposite side. The three medians meet at the centroid.
Incenter: The point where a triangle’s internal angle bisectors intersect, equidistant from its sides. It is the center of the circle tangent to all three sides.
Inscribed circle: A circle inside a polygon that is tangent to every side. Its center is the incenter, determined by equal distances to the sides.
Right triangle: A triangle with one interior angle equal to 90 degrees. Its circumcenter is the midpoint of the hypotenuse, while its orthocenter is the right-angle vertex.
Perpendicular bisector: A line perpendicular to a segment at its midpoint. The perpendicular bisectors of a triangle’s sides meet at the circumcenter.
Circumcenter: The point where a triangle’s perpendicular bisectors intersect, equidistant from its vertices. It is the center of the circle through all three vertices.

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