Incenter

The incenter is the common intersection of a triangle’s three internal angle bisectors. It is equidistant from all three sides and is the center of the triangle’s incircle.

Angle bisector: A ray or line that divides an angle into two equal angles. The incenter is defined by the intersection of the triangle’s three internal angle bisectors.
Compass-and-straightedge construction: A geometric construction using only an unmarked straightedge and compass. Two constructed angle bisectors locate the incenter exactly.
Angle bisector theorem: A theorem relating the lengths into which an angle bisector divides the opposite side to adjacent side lengths. It describes how triangle angle bisectors meet the opposite sides while locating the incenter.
Centroid: The common intersection of a triangle’s three medians. The centroid is defined by medians rather than angle bisectors.
Triangle: A polygon with three sides and three interior angles. The incenter is defined for a triangle and depends on its three interior angles.
Angle trisection: The division of an angle into three equal angles. Unlike angle bisection, the incenter requires no trisection; two bisectors suffice.
Gergonne point: The concurrence point of lines joining a triangle’s vertices to the incircle’s points of tangency. The incircle centered at the incenter supplies the tangency points used to define this point.
Circumcenter: The common intersection of a triangle’s perpendicular bisectors, equidistant from its vertices. The circumcenter is vertex-equidistant, whereas the incenter is side-equidistant.
Incircle: The circle inside a polygon that is tangent to each of its sides. The incenter is the center of the circle tangent to all three sides.
Inradius: The radius of a circle inscribed in a polygon. It equals the incenter’s perpendicular distance to each side of the triangle.

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