Angle bisector theorem

In a triangle, an internal angle bisector divides the opposite side into two segments proportional to the lengths of the adjacent sides.

Triangle: A polygon with three sides and three interior angles. The theorem relates the sides and angles of this basic geometric figure.
Triangle construction: The construction of a triangle from specified side lengths, angles, or other geometric data. The theorem locates the bisector's foot when the triangle's sides are known.
Angle: A geometric figure formed by two rays sharing an endpoint. The theorem begins with a triangle vertex angle divided into equal parts.
Perpendicular bisector theorem: A theorem stating that points on a segment's perpendicular bisector are equidistant from its endpoints. It concerns equal distances from endpoints, unlike the angle bisector theorem's proportional side division.
Internal angle bisector: A ray from a triangle's vertex that divides its interior angle into two equal angles. Its intersection with the opposite side creates the segments in the theorem.
Ceva's theorem: A criterion stating when three cevians of a triangle meet at one point, using ratios on opposite sides. Its product of side ratios can establish concurrence of triangle angle bisectors.
Internal division: The division of a segment into parts by a point lying between its endpoints. The bisector meets the opposite side at an interior point, producing two lengths.
Median (geometry): A segment from a triangle vertex to the midpoint of the opposite side. A median divides the opposite side equally, while an angle bisector generally does not.
Similar triangles: Triangles with equal corresponding angles and proportional corresponding side lengths. An auxiliary parallel line creates similar triangles that yield the side ratio.
Incenter: The point where the three internal angle bisectors of a triangle intersect. The theorem describes each bisector's division of the opposite side on the way to their common point.

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